Rayleigh scattering

Rayleigh scattering (

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. 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 of

In 1869, while attempting to determine whether any contaminants remained in the purified air he used for infrared experiments, John Tyndall discovered that bright light scattering off nanoscopic particulates was faintly blue-tinted.^[3][4] He conjectured that a similar scattering of sunlight gave the sky its blue hue, but he could not explain the preference for blue light, nor could atmospheric dust explain the intensity of the sky's color.

In 1871,

Small size parameter approximation

The size of a scattering particle is often parameterized by the ratio

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

where r is the particle's radius, λ is the

$${\displaystyle I_{s}=I_{0}{\frac {1+\cos ^{2}\theta }{2R^{2}}}\left({\frac {2\pi }{\lambda }}\right)^{4}\left({\frac {n^{2}-1}{n^{2}+2}}\right)^{2}\left({\frac {d}{2}}\right)^{6}}$$

where R is the distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh

$${\displaystyle \sigma _{\text{s}}={\frac {8\pi }{3}}\left({\frac {2\pi }{\lambda }}\right)^{4}d^{6}\left({\frac {n^{2}-1}{n^{2}+2}}\right)^{2}.}$$

Here n is the refractive index of the spheres that approximate the molecules of the gas; the index of the gas surrounding the spheres is neglected, an approximation that introduces an error of less than 0.05%.

The fraction of light scattered by scattering particles over the unit travel length (e.g., meter) is the number of particles per unit volume N times the cross-section. For example, air has a refractive index of 1.0002793 at atmospheric pressure, where there are about 2×10²⁵ molecules per cubic meter, and therefore 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).^[15] This means that 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

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^[16]

$${\displaystyle I_{s}=I_{0}{\frac {8\pi ^{4}\alpha ^{2}}{\lambda ^{4}R^{2}}}(1+\cos ^{2}\theta )}$$

$${\displaystyle I_{s}=I_{0}{\frac {\pi ^{2}\alpha ^{2}}{{\varepsilon _{0}}^{2}\lambda ^{4}R^{2}}}{\frac {1+\cos ^{2}(\theta )}{2}}}$$

Effect of fluctuations

When the

${\displaystyle \epsilon }$

${\displaystyle V}$

${\displaystyle {\bar {\epsilon }}}$

$${\displaystyle I=I_{0}{\frac {\pi ^{2}V^{2}\sigma _{\epsilon }^{2}}{2\lambda ^{4}R^{2}}}{\left(1+\cos ^{2}\theta \right)}}$$

${\displaystyle \sigma _{\epsilon }^{2}}$

${\displaystyle \epsilon }$

Cause of the blue color of the sky

The blue color of the sky is a consequence of three factors:

• the
  blackbody spectrum of sunlight
  coming into the Earth's atmosphere,
• Rayleigh scattering of that light off oxygen and nitrogen molecules, and
• the response of the human visual system.[18]

The strong wavelength dependence of the Rayleigh scattering (~λ⁻⁴) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths. This results in the indirect blue and violet light coming from all regions of the sky. The human eye responds to this wavelength combination as if it were a combination of blue and white light.^[18]

Some of the scattering can also be from sulfate particles. For years after large

$${\displaystyle \alpha _{\text{scat}}={\frac {8\pi ^{3}}{3\lambda ^{4}}}n^{8}p^{2}kT_{\text{f}}\beta }$$

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

• Rayleigh sky model
• Rician fading
• Optical phenomena
   – Observable events that result from the interaction of light and matterPages displaying short descriptions of redirect targets
• Dynamic light scattering – Technique for determining size distribution of particles
• Raman scattering – Inelastic scattering of photons by matter
• Rayleigh–Gans approximation
• Tyndall effect – Scattering of light by tiny particles in a colloidal suspension
• Critical opalescence
• HRS Computing – scientific simulation software
• Marian Smoluchowski – Polish physicist (1872–1917)
• Rayleigh criterion
   – Ability of any image-forming device to distinguish small details of an objectPages displaying short descriptions of redirect targets
• Aerial perspective – Atmospheric effects on the appearance of a distant object
• Parametric process – Interacting phenomenon between light and matter
• Bragg's law – Physical law regarding scattering angles of radiation through a medium

Works

• Strutt, J.W (1871). "XV. On the light from the sky, its polarization and colour". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (271): 107–120.
  doi:10.1080/14786447108640452
  .
• Strutt, J.W (1871). "XXXVI. On the light from the sky, its polarization and colour". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (273): 274–279.
  doi:10.1080/14786447108640479
  .
• Strutt, J.W (1871). "LVIII. On the scattering of light by small particles". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (275): 447–454.
  doi:10.1080/14786447108640507
  .
• Rayleigh, Lord (1881). "X. On the electromagnetic theory of light". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 12 (73): 81–101.
  doi:10.1080/14786448108627074
  .
• Rayleigh, Lord (1899). "XXXIV. On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 47 (287): 375–384.
  doi:10.1080/14786449908621276
  .

Wikimedia Commons has media related to Atmospheric Rayleigh scattering.

Further reading

• 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 978-0-12-218101-6
  .
• Chakraborti, Sayan (September 2007). "Verification of the Rayleigh scattering cross section".
  S2CID 119100295
  .
• 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 978-0-314-02779-5
  .
• Lilienfeld, Pedro (2004). "A Blue Sky History". Optics and Photonics News. 15 (6): 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