Rayleigh scattering

Rayleigh scattering [ reɪlɪ - ], named after William John Strutt, 3 Baron Rayleigh called elastic scattering of electromagnetic waves of particles whose diameter is small compared to the wavelength λ, ie at about the scattering of light by small molecules.

The scattering cross section σ of Rayleigh scattering is proportional to ω4 ( ω = angular frequency of the electromagnetic wave ). This is true not only for independently scattering particles, ie with particle distances greater than the coherence length of the radiation, but at a higher concentration of particles for the scattering on inhomogeneities of the refractive index by a random arrangement of the particles, for example, in gases or glasses. Blue light has a higher frequency ω than red and is therefore more dispersed. This effect is responsible for the sky blue during the day and for the dawn ( and sunsets ) of the sun on the horizon. For the same reason, the data is then transmitted through glass fibers with even longer wavelengths, infrared light.

Rayleigh scattering occurs because the incident light excites the electrons in the molecule and a dipole moment induced which resonates exactly as the incident electromagnetic radiation. The induced dipole moment now acts like a Hertzian dipole and emits light having the same wavelength as the incident light.


The cross section of the Rayleigh scattering for a single particle is derived from the oscillator model. In the limit of low frequency (compared to the natural frequency ) where:

Wherein the Thomson effect cross section (see Thomson scattering).

For a small ball n with diameter d and refractive index of the intensity I at the distance R of the scattered light at the angle

The incident wave intensity is assumed unpolarized. The scattered light is divided in the ratio 1: the polarization directions perpendicular and parallel to the scattering plane.

The blue or the red of the sky

The Rayleigh scattering explains why the sky appears blue. The wavelength of blue light, is approximately 450 nm, the red light around 650 nm Thus it follows for the ratio of the cross sections:

The picture shows the radiated power distribution of the sun, approximated by Planck's law of radiation from a surface temperature of 5777 K is shown in red. The spectral maximum is then at green light ( 500 nm wavelength). The spectral maximum of daylight, however, is partly due to the scattering effect described here, at 550 nm, the power distribution of the scattered light (blue curve) is obtained by multiplying with ω4. Accordingly, the maximum wanders far into the UV range. In fact, it is but in the near UV, in the sense that molecular absorptions at shorter wavelengths.

  • On the day when the sun is high in the sky, the light travels only a short distance through the atmosphere back. Only little blue light is scattered in other directions. Therefore, the sun appears yellow. From high-flying aircraft from the sun appears "white " because fewer missing blue light components.
  • The sum of all the scattered light makes the sky appear blue from all other directions. On the moon, where a dense atmosphere is absent, the sky appears black, however, also during the day.
  • When the sun is low the path of sunlight through the atmosphere is much longer. This allows most of the high-frequency light components (blue) is scattered away from the side, it remains predominantly light with long wavelengths remain and the perceived color of the sun shifts toward the red. This effect is further reinforced by additional particles in the air (for example, mist, aerosol, dust).

Intensity of the light attenuation by Rayleigh scattering

To calculate the thickness of the Rayleigh scattering quantitatively has to be considered that, within a coherency volume of the radiation emanating from the interfering molecules elementary waves, so that not intensities above formula, but scattering amplitudes have to be added. The particle density, below which this effect can be neglected for sunlight, is about 1/μm ³, seven orders of magnitude less than the relevant value for the atmosphere. The average density within the scattering volume element is irrelevant to the scattering effect, the density variations.

A result of the statistic is that the fluctuation amplitude of the particle increases with the square root of the number of particles. Since shorter wavelengths are scattered at finer structures correspondingly fewer particles ' sees ' this radiation stronger fluctuation amplitudes than longer wavelength radiation. At a fixed wavelength, the fluctuation amplitude is dependent on the square root of the particle density of the gas. The scattered intensity, however, depends quadratically on the fluctuation amplitude from, ie linearly dependent on the density. Overall applies to the attenuation of light in the atmosphere at normal incidence by Paetzold (1952 ):

This is the so-called extinction in astronomical size classes, the refractive index of air under normal conditions, the effective thickness of the atmosphere ( scale height, see barometric height formula), and the particle density of the air ( again under normal conditions). That the latter is in the denominator, is only an apparent contradiction to the above, as the term is proportional to density and is quadratic in the numerator.

From the extinction in turn is followed by the transmission, the ratio between the transmitted and incident light from the scattering layer intensity:

This is the usual in astronomy form of the Lambert- Beer law. In practice, it is also

Used with an optical depth. It is the simple conversion:

At an oblique incidence at a zenith angle of the effective layer thickness is approximately (with plane-parallel stratification ):

In the visual (550 nm) at normal incidence to happen about 90 % of the light, the atmosphere in the blue (440 nm) for about 80%. In shallow incidence at a zenith angle of 80 °, these shares are only at 60 % and 25 %. The already discussed redness of the light by the Rayleigh scattering is so clear.

In practice, the light is weakened by other scattering aerosol and dust particles ( see Mie scattering ) is significantly greater. After natural disasters such as volcanic eruptions, this additional extinction is particularly strong. So Grothues and Gochermann ( 1992) found after the eruption of Pinatubo in 1991 at La Silla ( one of the sites of the European Südobservatoriums ( ESO) ), at normal incidence in the visible light attenuation of 0.21 size classes (normal 0.13 size classes ). The transmission was thus reduced from 89% to 82%. In Blue, the extinction coefficient was increased from 0.23 to 0.31 magnitudes, that is, the transmission had fallen from 81% to 75%.