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. After the Rayleigh scattering the state of material remains unchanged, hence
Rayleigh scattering is also said to be 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.
For historical reasons, 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. 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". 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 ,
where r is its characteristic length 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 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 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. 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 ,
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
. 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. 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 means that shorter wavelengths are scattered
more strongly than longer 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
Reason for the blue color of the sky
The strong wavelength dependence of the scattering means that shorter wavelengths are scattered
more strongly than longer 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. 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 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 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. 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:
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. Tf is a fictive temperature,
representing the temperature at which the density fluctuations are "frozen" in the material.
In porous materials
λ−4 Rayleigh-type scattering can also be exhibited by porous materials. An example
is the strong optical scattering by nanoporous materials. 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 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
Further reading Pedro Lilienfeld, "A Blue Sky History.". 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