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Rayleigh scattering

Elastic scattering of electromagnetic radiation by particles much smaller than the wavelength; explains blue skies, red sunsets, and wavelength-dependent attenuation in optics.

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Overview

Rayleigh scattering is the elastic scattering of electromagnetic radiation by particles or assemblies that are much smaller than the incident wavelength. When light encounters such tiny scatterers, it is redirected without a change in frequency, and the efficiency of that redirection depends strongly on the wavelength. This wavelength dependence explains many common optical effects in nature and technology.

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Arctic Sunset
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Mechanism and characteristics

In the Rayleigh regime the scatterers act like induced dipoles driven by the incident field. Two widely cited qualitative rules are useful:

  • The scattered intensity increases rapidly for shorter wavelengths, approximately following an inverse fourth-power law with respect to wavelength. That is why blue light is scattered much more than red.
  • The theory applies when the particle size is much smaller than the wavelength; under those conditions the scatter behaves differently than larger-particle scattering such as Mie scattering.

The angular distribution and polarization of Rayleigh-scattered radiation reflect dipole emission patterns: scattered light can be partially polarized and its brightness varies with observation angle. The scattering strength also depends on particle composition and size: small changes in radius or refractive index produce large changes in scattering efficiency.

History and development

The phenomenon and its first theoretical treatment are commonly associated with Lord Rayleigh, who formulated conditions and consequences for small-particle scattering. Later developments in scattering theory broadened the framework and connected elastic Rayleigh scattering to inelastic processes studied by others. In particular, the molecular inelastic scattering now known as the Raman effect is related historically and conceptually but differs because it changes photon frequency.

Examples and practical importance

Rayleigh scattering is responsible for familiar optical phenomena: the daytime sky appears blue because shorter (bluer) wavelengths are scattered more strongly out of the direct sunlight; near sunrise and sunset the sunlight traverses a longer atmospheric path and most short wavelengths are removed, leaving reddened light. The same wavelength-dependent process affects remote sensing, astronomical observations, and the design of optical systems.

  • Atmospheric optics: sky color, twilight hues, and polarization patterns used by animals and navigation aids.
  • Optical communications and fibers: intrinsic scattering contributes to attenuation, especially at shorter wavelengths.
  • Instrumentation: Rayleigh scattering measurements help estimate particle sizes, concentrations, and purity in gases and liquids, and are exploited in LIDAR and laboratory diagnostics.

Rayleigh scattering should be distinguished from a few adjacent concepts. When scatterers have sizes comparable to the wavelength, Mie scattering governs the angular and spectral behavior and often produces white or forward-peaked scattering. In contrast to Rayleigh's elastic process, Raman-type scattering is inelastic and yields frequency-shifted light that reveals molecular vibrational information. More generally, the simple Rayleigh limit breaks down if the material or particle geometry becomes complex, or if absorption and multiple scattering are significant.

For concise introductions and visual explanations see summaries on radiation and atmospheric optics, and refer to technical treatments for mathematical formulation. The basic condition for applicability can be remembered: Rayleigh scattering applies when an incoming wave interacts with a very small object and the resulting scattering pattern and intensity depend strongly on the incident light properties and its wavelength, while the qualitative act of redirection is similar to the way a tiny dipole scatters electromagnetic waves.

Rayleigh scattering is the cause of aerial perspective
Rayleigh scattering is the cause of aerial perspective
Rayleigh scattering causes the blue hue of the sky during the day and the red hue of the sun when it sets or rises.
Rayleigh scattering causes the blue hue of the sky during the day and the red hue of the sun when it sets or rises.
The waxing crescent moon also appears reddish when it is only a few degrees above the horizon. The moonlight now reaches the observer only after a longer passage of more than 200 kilometres through the Earth's atmosphere.
The waxing crescent moon also appears reddish when it is only a few degrees above the horizon. The moonlight now reaches the observer only after a longer passage of more than 200 kilometres through the Earth's atmosphere.
Rayleigh-scattered light is polarized, especially strongly at scattering angles of 90°. The left image is taken without, the right one with a polarizing filter in blocking direction.
Rayleigh-scattered light is polarized, especially strongly at scattering angles of 90°. The left image is taken without, the right one with a polarizing filter in blocking direction.

Cross section

The effective cross section σ \sigma of Rayleigh scattering for a single particle is obtained from the oscillator model. In the limiting case of low frequencies (compared to the natural frequency, ω \omega \ll \omega _{0}) holds:

\sigma (\omega )\approx \sigma _{{\mathrm {Th}}}{\frac {\omega ^{4}}{\omega _{0}^{4}}}

Where σ \sigma _{{\mathrm {Th}}}=0{,}665\cdot 10^{{-24}}\,{\mathrm {cm}}^{2}the Thomson effective cross section. The angular distribution and polarization is that of a dipole in the direction of the incident wave.

If there are several particles in the coherence volume, the scattered waves interfere. When there are many particles per coherence volume, spatial variations of the refractive index act as scattering centers. Thus, for a sphere with diameter {\displaystyle d\ll {\tfrac {\lambda }{2\pi }}={\tfrac {1}{k}}}and refractive index n_{2}in a medium n_{1}is the scattering cross section:

{\displaystyle \sigma ={\frac {8\pi d^{6}k^{4}}{3}}\left({\frac {\left({\frac {n_{2}}{n_{1}}}\right)^{2}-1}{\left({\frac {n_{2}}{n_{1}}}\right)^{2}+1}}\right)^{2}.}

Milky opal is red-orange in transmission because density inhomogeneities scatter the blue light laterally.
Milky opal is red-orange in transmission because density inhomogeneities scatter the blue light laterally.

The blue respectively the red of the sky

Rayleigh scattering explains why the sky appears blue. The wavelength of blue light, λ \lambda _{{\mathrm {blau}}}, is about 450 nm, and that of red light is about 650 nm. Thus it follows for the ratio of the effective cross sections:

{\displaystyle {\begin{aligned}{\frac {\sigma _{\text{blau}}}{\sigma _{\text{rot}}}}&={\frac {\frac {1}{\lambda _{\text{blau}}^{4}}}{\frac {1}{\lambda _{\text{rot}}^{4}}}}=\left({\frac {650\,{\text{nm}}}{450\,{\text{nm}}}}\right)^{4}\approx 4{,}4\end{aligned}}}

In the picture, the radiated power distribution of the sun, approximated by Planck's radiation law from a surface temperature of 5777 K, is drawn in red. The spectral maximum is then at green light (500 nm wavelength). The spectral maximum of daylight, on the other hand, is at 550 nm, partly due to the scattering effect described here. The power distribution of the scattered light (blue curve) is obtained by multiplying by ω4. According to this, the maximum moves far into the UV range. In fact, however, it lies in the near UV, since molecular absorptions are added at shorter wavelengths.

  • During the day, when the sun is high in the sky, the light only travels a short distance through the atmosphere. In the process, only a little blue light is scattered in other directions. This is why the sun appears yellow. From high-flying aircraft, the sun appears "whiter" because less blue light has been scattered away.
  • The sum of all scattered light makes the sky appear blue from all other directions. On the moon, however, where there is no dense atmosphere, the sky appears black even during the day.
  • When the sun is low, the path of sunlight through the earth's atmosphere is much longer. As a result, a large proportion of the high-frequency light components (blue) are scattered to the side, light with long wavelengths predominantly remains and the colour impression of the sun shifts towards red. This effect is further enhanced by additional particles in the air (e.g. haze, aerosols, dust). However, Chappuis absorption is responsible for the blue colouring of the sky at the zenith after sunset, which is hardly noticeable at higher sun elevations.
Power distribution of scattered sunlight
Power distribution of scattered sunlight

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