# Scattering – overview – case of the air molecules

Scattering is a physical phenomenon specific to the interactions between radiation and matter. It is present at all wavelengths, even if the intensity of the phenomenon can vary with the wavelength. Let be an incident radiation composed of photons. Any scattering body like a particle or a molecule located in the path of a photon deflects the latter, more or less strongly. If the photon takes a direction other than that of incidence, then the energy in the incident direction decreases. In this way, the scattering phenomenon subtracts energy from the incident radiation.

As in the case of absorption, a scattering coefficient *b* can be defined. If Fis the incident flux, the flux *F'* transmitted over a length / in the scattering medium is:

The dimension of this coefficient *b* is the inverse of a length, *b* can be defined for a wavelength, a range of wavelengths, or the entire spectrum; it is called spectral scattering coefficient, scattering coefficient for an interval, or total scattering coefficient, respectively.

The way in which a body scatters energy according to the direction is called the scattering pattern or the phase function. The phase function gives the probability that a photon is scattered in a given direction. If the scattering is isotropic, the energy is scattered equally in all directions. The phase function and the scattering pattern depend on the body itself, its nature, its shape, its size, and its surface properties as well as on the wavelength.

Figure 4.3 is a schematic view of a possible scattering pattern in the case where the size of the scattering body is much smaller than the wavelength of the incident radiation. An incident radiation (black arrow on the left) can be scattered in all directions, partly indicated by the dotted arrows. The scattering pattern is shown in gray line. The further it is from the scattering body represented by the small black circle in Figure 4.3, the greater the probability that the photon is scattered in this direction.

*Figure 4.3* Schematic view of a possible scattering pattern in the case where the scattering body (small black circle) is much smaller than the wavelength of the incident radiation shown by the solid arrow.

*Figure 4.4* Schematic view of possible scattering patterns in the case where the size of the scattering body (small black circle) is comparable to or slightly greater than the wavelength of the incident radiation shown by the solid arrow.

In this diagram, the scattering pattern is almost an ellipse: The probability that the photon is scattered forward, i.e., in the incident direction, or backward, is greater than the probability of scattering at 90° or -90°, i.e., perpendicular to the incident direction. This pattern is typical of scattering patterns for gas molecules in the atmosphere.

Two other scattering patterns are given in Figure 4.4 for particles larger than the molecule in the previous case. In both diagrams, the scattering patterns are very elongated forward, which means that the probability is very high for the photons to continue in the incident direction. In the upper diagram, photons are unlikely to be scattered in perpendicular directions; more often they will be scattered mainly forward or backward. As the size of the particle increases, the probability that the photons are scattered forward generally increases, with scattering patterns that may have complex shapes as shown in the lower diagram. The scattering patterns shown here are only schematic representations. In reality, the particles are not smooth spheres and the scattering patterns are more complicated and asymmetrical.

The atmosphere contains many molecules and particles. Frequently, one of these scattering bodies scatters photons that have already been scattered by other bodies, and so on. This multiple scattering is illustrated in Figure 4.5. The two lines frame the incident radiation beam, formed by parallel rays. Photons are scattered several times in this beam. They can remain there or leave it, or even reintegrate it after multiple scattering outside the beam.

Scattering by bodies of comparable size or slightly greater than the wavelength of the incident radiation can be described by Mie’s law.^{[1]} The latter describes the scattering by a homogeneous sphere. Particles in the atmosphere are not homogeneous perfect spheres, and Mie’s law does not apply perfectly. However, it offers a first approximation of the scattering by aerosols, these particles, solid or liquid, suspended in the atmosphere, often located in the lower layers of the atmosphere, from the ground up to about 2 km altitude.

*Figure 4.5* Illustration of multiple scattering within an atmospheric column.

When the scattering bodies are much smaller than the incident wavelength, scattering can be described by Rayleigh’s law.^{[2]} This is the case for air molecules and other gases; this is why a clean and dry atmosphere that contains only such molecules is sometimes called the Rayleigh atmosphere. According to this law, the variation of the scattering coefficient is in 2 , where *X* is the wavelength. This means that the smaller the wavelength, the more important the scattering.

Scattering by molecules is well marked at wavelengths less than about 700 nm. The influence of this scattering can be seen in Figure 4.2 showing an example of the spectral distribution of the fraction of solar flux transmitted by the atmosphere. This results in a rounded shape of the distribution up to 700 nm. Incident radiation loses more energy in its downwelling path to the ground at short wavelengths than at long ones.

In the visible part of the solar spectrum, blue photons are more scattered than those with longer wavelengths, the least scattered being red photons. As shown in Figure 4.3, the scattering pattern for air molecules is almost an ellipse; i.e., the probability of scattering is noticeable for any direction, even if forward or backward scattering exhibits the highest probabilities. As a result, blue photons are scattered in all directions, much more than longer photons. They form an important part of the radiances coming from all directions of the sky vault, except the direction of the sun. When the sky is very clear, the portion of the sky far from the direction of the sun appears dark blue to the observer because the radiances mainly include blue photons. As water vapor increases, absorption and scattering increase at all wavelengths that are more multiscattered than in the case of a dry atmosphere. The sky vault then appears rather white.