Rayleigh scattering

Rayleigh scattering causes the blue hue of the daytime sky and the reddening of the sun at sunset
Rayleigh scattering is more dramatic after sunset. This picture was taken about one hour after sunset at 500m altitude, looking at the horizon where the sun had set.
Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light.

Rayleigh scattering (named after the British physicist Lord Rayleigh) is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light, which may be individual atoms or molecules. It can occur when light travels in transparent solids and liquids, but is most prominently seen in gases. Rayleigh scattering is a function of the electric polarizability of the particles.

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).

Contents

Small size parameter approximation

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 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 is dependent 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.

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.

\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, 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.

Rayleigh scattering from molecules

The beam of a 5 mW green laser pointer is visible at night due to Rayleigh scattering and airborne dust.

Rayleigh scattering from molecules is also possible. An individual molecule does not have a well-defined refractive index and diameter. Instead, a molecule has a 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[1]

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).[2] 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), in the atmosphere, means that shorter blue wavelengths are scattered much more readily than longer red wavelengths, and so one sees blue light coming from all regions of the sky. Direct radiation (by 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 a small size parameter.

Reason for the blue color of the sky

Light from the sun that does not happen to be traveling toward our eyes scatters off molecules and other small particles in the atmosphere. As previously explained, 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 of green and especially red light. Although the most strongly scattered wavelength of visible light in the sky is violet, limitations in the sensitivity of human eyes to shorter wavelengths of light means that the violet light in the sky is detected only weakly by our eyes. As a result, the sky is perceived as blue despite the fact that it is chiefly violet [3]. Conversely, glancing toward the sun, the colors that were not scattered away -- the longer wavelengths such as red and yellow light -- are visible, giving the sun itself a slightly yellowish hue. Viewed from outer space, 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 therefore increased, radiating even more of the sun's shorter wavelength (violet and blue) light in different directions. The remaining unscattered light that is received by an observer is mostly of a longer wavelength and therefore appears to be red.

Rayleigh scattering primarily occurs through light's interaction with air molecules. Or, which is the same thing, from a purely "optical", macroscopic point of view, we can say that blue sky comes 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 reflect light in random directions. Smaller regions fluctuates more than larger ones, and, since short wavelenghts are "disturbed" by small regions more than longer wavelenghts, they are scattered more.

Some of the scattering can also be 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.

In locations with little light pollution, the moonlit night sky is also blue, for the same reasons that the sky is blue during the day (moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the moon). We do not perceive the moonlit sky as blue because at low light levels, human vision comes mainly from rod cells that do not produce any color perception.

Rayleigh scattering 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 gives rise to energy loss due to the scattered light, with the following coefficient[4]:

\alpha_{scat} = \frac{ 8 \pi^3}{3 \lambda^4} (n^8p^2) (k T_f)\beta

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

Rayleigh scattering in porous materials

Rayleigh scattering in glass: it appears blue from the side but orange light shines through.[5]

λ−4 Rayleigh-type scattering can also be exhibited by porous materials. An example is the strong optical scattering exhibited by nanoporous materials [6]. In [6], the strong contrast in refractive index between pores and the solid of sintered alumina results in very strong scattering (light completely changing direction, on average, each 5 micrometers), and 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

References

  1. ↑ Rayleigh scattering at Hyperphysics
  2. ↑ 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).
  3. ↑ "Why is the sky purple?" Colorado State University. http://littleshop.physics.colostate.edu/docs/CMMAP/tenthings/SkyPurple.pdf
  4. ↑ K.Rajagopal, Textbook on Engineering Physics, PHI, New Dehli 2008, part I, Chapt. 3
  5. ↑ http://www.webexhibits.org/causesofcolor/14B.html
  6. ↑ 6.0 6.1 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. 
  • Chakraborti, Sayan (September 2007). "Verification of the Rayleigh scattering cross section". American Journal of Physics 75 (9): 824−826. 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. 

External links