Scattering

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The Colours of the Sky

Dietrich Zawischa

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In the air, part of the sunlight is scattered. The small particles (molecules, tiny water droplets and dust particles) scatter photons the more, the shorter their wavelength is. Therefore, in the scattered light, the short wavelengths predominate, the sky appears blue, while direct sunlight is somewhat yellowish, or even reddish when the sun is very low. Goethe believed this to be the basic phenomenon to generate colours (“Urphänomen”).

Due to the scattering of light the wood in the background is seen like behind a bluish veil (“aerial perspektive”).

The fraction of the light which is deviated by scattering increases with increasing path length, so that at sunset the shorter wavelengths are depleted in direct sunlight and the sun appears orange or red, depending on the amount of haze or dust in the air.
sunset

Just before sunset on a hazy day. The sun is about 0.8º above the horizon. On the right hand side: enlarged detail.

More images with sunsets

After sunset on a clear day. In the west the sky is brightest near the horizon (left), in the east it is darkest there due to the earth's shadow. Above the shadow zone there is a slight pink reflection of the setting sun's red light.

Lunar Eclipse

The colours of sunrise and sunset sometimes even show themselves on the moon. When during a lunar eclipse the earth shades the moon, some light is scattered into the shadow region by the earth's atmosphere. This light is reddish, as shorter wavelength light is largely scattered to other directions. As viewed from the moon, the earth's atmosphere would be seen as a shining border, red at the inner side if there are no clouds, and becoming pale and bluish outwards.

Partial lunar eclipse on August 16, 2008, photograph taken at 23:15 CEST

Lunar eclipse, February 21, 2008

Rayleigh Scattering

Now look at a single “air particle”, i.e. a nitrogen or an oxygen molecule. Fine details are not relevant now, we adopt a simple model of positive electric charge (containing almost all of the mass) surrounded by a cloud of negative charge. The dimension of this assembly is of the order of tenths of a nanometer, and thus several thousand times smaller than the wavelength of visible light.
If the molecule is exposed to an electric field, the centroids of the positive and the negative charges are somewhat pulled apart, the molecule becomes a minute electric dipole.
As the electrons have much smaller mass than the nuclei, the motion of the nuclei may be ignored at all, and the model for the molecule is reduced to a negatively charged small mass bound by some elastic force (of electrical origin) to an infinitely heavy nucleus.
If there is incident light with a certain wavelength and corresponding frequency, a periodic force acts on the elastically bound charge which induces forced oscillations. It is important that the frequencies of visible light are much smaller than the resonance frequencies of the molecules. From this one can conclude that the oscillation amplitudes are independent of the exciting frequency to a first approximation.
The molecule which we consider thus becomes an oscillating dipole, oscillating with the same frequency as the incoming wave, thus acts like a tiny antenna and emits electromagnetic radiation.
If the centroid of the negative charge −q is separated from that of the positive charge +q by a distance x₀, then the product q.x₀, which is called the dipole moment, is a suitable measure for the strength of the dipole (because the resulting field depends only on that quantity as long as the separation distance x₀ is very small compared to other relevant distances). For not too strong external fields, the dipole moment is proportional to the external field strength; the dipole moment divided by the field strength is called polarizability. The polarizability is the physical quantity characterizing the molecule which is relevant here, and which, for the cases of interest here, is almost independent of the frequency.
The calculation of the radiation emitted by an oscillating dipole can be found in textbooks on electromegnetism. The result is that the emitted radiative power is proportional to the square of the dipole moment and to the fourth power of the oscillation frequency. This is a consequence of the smallness of the dipole compared with the wavelength of the radiation. For a transmitting aerial (antenna) the optimal size is of the order of about half the wavelength, for maximum transmitting power. The fraction “dipole-size over wavelength” is more favourable for short wavelengths. Thus, more light with short waves is scattered than with long waves, inversely proportional to the fourth power of the wavelength. Light with 450 nm wavelength (blue) is scattered more than four times stronger than light of 650 nm (red).

This causes the sky to be blue.

computed
colour of the sky

Perhaps a calculation which is not explicitely shown is not very convincing. The result, however, is quite plausible, as the following example may show:
It is possible to excite water waves by moving one's hand periodically up and down slightly below the surface. If this is done rapidly enough, clearly visible waves are generated. 20 to 30 cm (8 to 12 in) distance from crest to crest can easily be produced. To excite longer waves, the hand must be moved more slowly, but then soon there will be no noticeable waves any more. The more slowly the hand is moved, the less power goes into the waves produced, the hand is too small to produce long waves. (If, on the contrary, a large island moves a little bit in an earthquake, very long waves may be excited.)

Now consider a gas. Each single molecule of it scatters light. In principle, to find out how much scattered light reaches a certain point in space, one has to add the field strengths of all the waves coming from the different scattering centres, and the intensity is proportional to the square of the sum so obtained. There is interference (i.e. spatially changing mutual enhancement, weakening or cancellation) among the waves from the different scatterers. As these are distributed randomly, and, moreover, in permanent motion, the interference terms change very rapidly and cannot be observed. One can only see spatial and time averages, and under these circumstances, the intensities (and not the field strengths) coming from the single scatterers add up. Thus, the intensity of the scattered light is inversely proportional to the fourth power of the wavelength (Rayleigh scattering).

This power law holds as long as multiple scattering is negligible. If, however, the light has to pass a longer distance through the scattering medium, multiple scattering will occur. Multiple scattering tends to wash out the strong wavelength-dependence, so that on a bright day the blue colour of the sky is much less saturated near the horizon than higher up.

Next we consider a small droplet which consists of N molecules of water and is assumed to be very small compared to the wavelengths of visible light. To simplify things, we assume that each of the molecules feels the field of the incident wave. In this crude approximation, the whole droplet behaves like a single molecule with an N-times larger polarizability, and the scattered wave has N²- times the intensity than that of a single molecule. As long as the droplet is small compared to the wavelengths, the short waves are scattered more than the longer ones.

Larger droplets

With increasing droplet size the situation changes: the scattered waves coming from different parts of the drop interfere and partially extinguish each other. For large drops, the scattered waves coming from inside cancel each other, and what remains is interpreted as reflected and refracted waves. In this case (we speak of Mie scattering) the spectral composition of the scattered light depends on the scattering angle. In dense clouds, this dependence is averaged out by varying drop sizes and multiple scattering and the clouds appear white or grey. Thus, if the scattering particles are larger than the wavelengths of light, the light is scattered much more, but long and short waves are affected equally.
In bulk (nonabsorbing) matter of constant density, the scattered waves cancel each other in all directions except the direction of propagation. Instead of scattering, the polarizability leads to a change of the wavelength which is accounted for by the introduction of a refractive index.

Tyndall Effect In clear water, light is scattered only to a negligible amount. Scattering occurs again, if small particles of different polarizability are suspended (Tyndall effect). Mastic (or some other kind of resin) dissolved in alcohol and added to water yields a turbid medium to observe this effect. The size of the suspended particles is decisive: if they are small compared to the wavelengths of light, colours are seen; if they are larger, the suspension appears just whitishly turbid. The images to the right show a suspension of spruce resin in water, lit by white light from a LED. At the very right: view through the liquid column on the lamp.
Opal, an amorphous form of silica, is made up of submicroscopic silica spheres closely packed, the spaces between the spheres containing water or water-rich silica. Tyndall scattering at these inhomogeneities leads to bluish colour in reflected light, while in transmitted light it looks yellowish. In precious opal, macroscopic domains consist of spheres of uniform size, so that the inhomogeneities form regular lattices, as can be seen in the REM pictures supplied by the Mineral Spectroscopy Server of the California Institute of Technology: opal_gem, opal-beads. Such regular lattices cause colourful reflections of light. Right: an opal from Ethiopia, length 22 mm

Turbid, milky or opalescent glass is produced by adding fluorides (cryolite, Na₃Al F₆, sodium hexafluoroaluminate), calcium phosphate (bone ash) or tin dioxide to the smelting charge. By this, submicroscopically small crystals finely dispersed in the glass are obtained which scatter the light due to their different polarizability. Cryolite glass (image to the right) looks bluish before a dark background and orange-yellow in transmitted light.
Scattering of light (Tyndall effect and, closely related, Rayleigh scattering) can thus be observed quite frequently. The iris of the human eye does not contain any blue pigment or dye. The turbid front layer, if it contains no or only little melanin, appears blue in front of the dark back layer due to the preferred scattering of light with short wavelengths. Blue eyes are probably the best known example of Tyndall scattering.

Goethe recommends to soak the bark of chestnut trees in water to obtain a suitable turbid fluid looking blue in front of a dark background. But the bark of horse chestnuts contains aesculin which shows blue fluorescence like the optical brighteners which are commonly added to laundry detergents. The blue fluorescence should not be misinterpreted as Tyndall scattering.

Absorption

The presupposition that the “air molecules” do not absorb visible light must be emphasized. Often there are no dramatic changes in the Tyndall effect due to absorption: cigarette smoke looks bluish grey before a dark background but yellowish grey before a light one. But if there is strong absorption (resonance), much more complicated effects are possible: Tiny spherules of gold dispersed in glass (“colloidal gold solution”) produce the red colour of ruby glass. This is discussed in the chapter on pigments, in the section on metals.

Density Fluctuations

Rayleigh scattering has been discussed for the case of rarefied gases, in that case the intensities of the scattered waves had to be summed up. At atmospheric pressure, a cube with edges 400 nm long (corresponding to the shortest visible wavelengths) contains 1.7·10⁶ molecules. Are the conditions for independent scattering on the single molecules still valid or is this situation more similar to a uniform density where the scattering only leads to a refractive index? This question has been answered in the beginning of the 20th century by Einstein and Smoluchowski who found that scattering by the density fluctuations of a gas yields exactly the same as scattering on independent molecules.

Of course, the waves scattered by different particles interfere; but if the positions of the particles are randomly distributed, in the case of large numbers of scatterers, the interference terms are averaged to zero. Thus the intensity of the scattered light is just the sum of intensities from the individual scatterers.

Blue and green colours of animals

In the animal kingdom, there are many examples of non-iridescent blue colour which is not produced by blue pigments. These are caused by layers of tissue with small-scale density variations (e.g. by suspended nanoparticles with high refractive index), on top of tissue coloured black by melanins. However, recent investigations have shown that the spectrum of the blue remitted light does not conform to the well known Rayleigh shape (inversely proportional to the fourth power of the wavelength), due to the fact that the positions of the scattering centres are not completely random; instead, there are short-range correlations (Prum et al., 1999a, 1999b, 2004a, 2004b). Thus, it is argued, the non-iridescent blue should rather be considered as a structural colour.

Left: Feather of an Eurasian jay. Image width 4 cm. Right: enlarged detail.

We speak of Tyndall scattering in those cases where the scatterers are distributed with low density in a larger volume. To achieve noticeable scattering already in a thin layer, the particle density must be much higher. If the distribution were truely random, clots would form. Therefore, the organism tries to keep a certain minimum distance between the scattering particles. This does not lead to long-range order, but makes the distribution much more uniform. This is illustrated by the following stereo images.

Left pair: random distribution, right: statistical distribution with minimum distance between particles. Click to enlarge; cross-eyed viewing is recommended.

A tentative model computation

A small spherical domain with radius 4 μm is assumed to contain small Tyndall scatterers which are (a) randomly distributed or (b) must have a distance of at least 200 nm to their nearest neighbours. The coordinates are obtained from a random number generator. The scattered intensities show strong statistical fluctuations; to reduce these, average values over a larger number of similar systems are taken. Multiple scattering is ignored.

The graphs below show the intensities multiplied by the fourth power of the wavelength for the scattering angle of 120° (arbitrary units).


(a) Tyndall scattering by about 15700 randomly placed particles in a spherical volume of 4 μm radius. An average has been taken over 200 systems.		(b) Scattering by likewise about 15700 particles obeying a minimum distance of 200 nm to their neighbours, average over 80 samples.
In the case of Tyndall scattering, for “infinitely many” particles the quantity shown would yield a horizontal straight line. It is seen that the correlations change the colour of the scattered light towards a more sarurated blue, as the longer wavelengths are supressed. The results are in agreement with the experimental findings of Prum et al.
Green colour is achieved by adding a transparent top layer of yellow cells. Their pigment (mostly a carotenoid) acts like a filter and quenches the short waves. This is modeled in the adjacent figure. The blue line is the scattered light without the yellow layer, the same as in the right image above. The thin yellow line is the transmittance of a yellow filter (slightly idealized), and the green line is obtained by applying the filter to the spectral distribution given by the blue line. The colour without and with the yellow layer is also shown.

Hyla arborea, European tree frog
(photograph: User:Ineptus)

The green colour of amphibia and reptiles is produced in that manner. Different chromatophore cells are arranged in layers. Yellow chromatophores are the outermost, followed by guanophores (or iridophores) containing the suspension of guanine crystals on top of a dark layer of melanophores. (See: “Nature's palette” by Margareta Wallin.)

Two parrots, photograph: RoFra,
license CC BY 3.0

Birds also make use of light scattering to produce green and blue colour, if these colours are not iridescent. How it is achieved is described in the article “Die Gefiederfarben der Vögel” (in German, sorry): The branches (barbs) of parrots' feathers which are fused to the feather's shaft consist of dark marrow surrounded by giant cells containing finely dispersed melanine particles. Due to their smallness and low concentration scattering outweighs absorption, and in front of the dark marrow they look blue. The outermost cell layer is transparent, colourless (blue feathers) or yellow, containing some carotenoid colourant (green feathers). The small barbules attached to the barbs are colourless or dark if they contain melanin.

Ischnura elegans (van der Linden)
Blue-tailed Damselfly, male

Some damselflies and dragonflies are decorated with non-iridescent, light blue colour. It is supposed that there the cells of the epidermis below the transparent cuticula contain a suspension of submicroscopic particles while the layer behind is coloured black by melanine (after Sternberg & Buchwald (1999): Die Libellen Baden-Württembergs, Band 1. (© Ulmer Verlag, Stuttgart), see “Körperbau der Libellen”).

Guanophores in front of a dark background layer are seen blue like the blue sky as long as the suspended particles are small compared to the visible wavelengths and thus scatter the shorter waves much more than the longer ones. White light is remitted if the crystals are larger, like the droplets in clouds. (Iridescence is also possible if the crystals form a regular lattice or are platelike and oriented.)

Structural colours produced by regular nanostructures like stacks of transparent layers (beetles, hummingbirds, butterflies) or space-lattice like structures (peacocks' and pheasants' feathers, butterflies) exhibit the whole spectrum of bright colours, while the irregular distribution of small scatterers yields only sky-blue hues. This is kind of in-between of Tyndall scattering and structural colour.

Appendix: Colour Fidelity and White Balance

Is it possible to assess the colour of the sky by photographs?
Below there are three pictures, taken within few seconds and showing the same part of the cloudy sky. The pictures differ by the pre-set white balance.

Left: setting “Daylight”
Middle: setting “Auto” (automatic white balance)
Right: setting “Cloudy”.

The picture in the middle is closest to the visual impression; and this is just what one expects from a photograph. The automatic white balance works well under most circumstances. This, however, does not answer the question of the “true” colour of the sky and the clouds and corresponds to our visual faculty which does not make absolute colour measurements either.
Second example:

Autumn leaves and blue, cloudless sky. Automatic white balance. Immediate comparison (looking alternatingly out of the window and at the screen) attests optimal reproduction of the colours.

The same autumn leaves one day later with overcast sky and drizzle.
Left: automatic white balance, right: setting “Cloudy”.
Here now, the right picture is closer to the visual impression. This could hardly have been decided without immediate comparison.

As the photographic camera does not get exactly the same information which is used by our eyes and brain to establish white balance, and since e.g. at cloudy weather, the light certainly is not always of the same colour, some careful subsequent treatment may be necessary.

Back to the index page “the origins of colour”

Continue with “diffraction and interference” (among otghers: iridescent clouds)

Continue with rainbows and other atmospheric phenomena

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