KAHIBARO
Table of Contents
Elastic scattering by bound electrons
Rayleigh scattering is the elastic scattering of a photon by an atom, molecule, or other bound collection of charges. Elastic means that the photon changes direction, but not its energy. Its wavelength stays the same.
In matter, a photon can interact in several ways. In the Rayleigh process, the photon does not ionize the atom and does not transfer enough energy to free an electron. Instead, the electric field of the incoming electromagnetic wave makes the bound electrons oscillate slightly. These oscillating charges then re-radiate electromagnetic waves. The result is a scattered photon with the same energy as the incident photon, but usually moving in a different direction.
This process is most important when the photon energy is low enough that the electrons remain bound during the interaction. It is therefore distinct from the photoelectric effect, Compton scattering, and pair production, which are treated elsewhere.
Rayleigh scattering is an elastic photon interaction.
$$E_{\text{after}} = E_{\text{before}}$$
Only the photon direction changes significantly, not its energy.
Physical picture
A simple way to picture Rayleigh scattering is to think of the electric field of light acting on the electron cloud of an atom. The charges are not free, they are attached to the atom by electromagnetic forces. The field causes a tiny oscillation of this charge distribution. Any accelerating charge emits radiation, so the atom re-emits part of the incident wave in different directions.
Because the charges are bound, the atom usually returns to the same internal state after the interaction. That is why no net energy is taken from the photon. The scattered radiation is therefore coherent with the incident radiation in the sense that the process preserves photon energy.
For very small particles compared with the wavelength, this same idea explains why short wavelength visible light is scattered strongly by tiny molecules in the atmosphere.
Dependence on wavelength
A key feature of Rayleigh scattering is its strong dependence on wavelength. For scatterers much smaller than the wavelength, the scattered intensity varies approximately as
$$I \propto \frac{1}{\lambda^4}$$
This means shorter wavelengths are scattered much more strongly than longer wavelengths. Blue light has a shorter wavelength than red light, so it is scattered more strongly in air. This is the basic reason the sky appears blue.
For Rayleigh scattering by particles much smaller than the wavelength,
$$I \propto \frac{1}{\lambda^4}$$
Shorter wavelength means much stronger scattering.
This strong wavelength dependence is one of the most famous signatures of Rayleigh scattering. It applies best when the scattering centers are much smaller than the wavelength of the radiation.
Angular distribution
Rayleigh scattered light is not emitted equally in all directions. For unpolarized incident light, the scattered intensity often follows an angular dependence of the form
$$I(\theta) \propto 1 + \cos^2\theta$$
where $\theta$ is the scattering angle measured from the incident direction. This means forward and backward scattering are stronger than scattering at right angles, although the exact form can depend on the structure of the scatterer.
At $90^\circ$, the scattered light can become strongly polarized. This is another important consequence of the oscillating dipole picture.
Rayleigh scattering in radiation physics
In nuclear and radiation physics, Rayleigh scattering is one of the photon interaction mechanisms in matter. It contributes to attenuation because photons are removed from the original beam direction, even though their energy is unchanged.
Suppose a narrow photon beam enters a material. Some photons pass straight through, while others are scattered out of the beam. A detector placed directly behind the material may count fewer photons. Part of this loss can be due to Rayleigh scattering.
This means Rayleigh scattering affects beam transmission and image formation, especially at lower photon energies. However, because there is no energy loss in the photon itself, it does not contribute to energy deposition in the same way as absorption processes do.
Comparison with other photon interactions
Rayleigh scattering is easy to confuse with other photon processes, so it helps to compare them directly.
| Interaction | Photon energy after interaction | Electron freed? | Main effect |
|---|---|---|---|
| Rayleigh scattering | Same | No | Direction changes |
| Photoelectric effect | Photon absorbed | Yes | Full absorption |
| Compton scattering | Reduced | Yes, or effectively transferred to electron | Direction and energy change |
| Pair production | Photon disappears | No ordinary electron ejected, creates $e^-e^+$ pair | New particles created |
The special feature of Rayleigh scattering is that it is elastic and involves bound electrons acting together within the atom.
Cross section and atomic dependence
The probability of Rayleigh scattering is described by a scattering cross section. In general, this probability depends on photon energy, scattering angle, and the atomic structure of the material.
For atoms with more electrons, coherent scattering can become stronger because the scattered waves from different bound electrons can add together. This often makes Rayleigh scattering more noticeable in materials with higher atomic number, especially for low energy photons.
A full quantitative treatment uses atomic form factors, which describe how the electron cloud distribution affects scattering. For beginners, the important idea is that the atom is not a point, and its internal charge distribution matters.
Rayleigh scattering depends on the bound electron structure of the atom.
It is not just a collision with one free electron.
Everyday example, the blue sky
A classic example is sunlight traveling through Earth’s atmosphere. Air molecules are much smaller than the wavelength of visible light, so Rayleigh scattering applies well. Since blue and violet light have shorter wavelengths than red light, they scatter more strongly.
Even though violet scatters more strongly than blue, the sky appears mainly blue because the Sun emits less violet light, and human eyes are less sensitive to violet.
At sunrise and sunset, sunlight travels through a longer path in the atmosphere. Much of the blue light is scattered out of the direct line of sight, so the transmitted sunlight looks redder.
Limits of the Rayleigh description
Rayleigh scattering is most accurate when the scatterers are much smaller than the wavelength. If the particles become comparable in size to the wavelength, the scattering behavior changes and a more general treatment is needed. For larger particles, the wavelength dependence is weaker, which is why clouds, made of larger water droplets, usually look white rather than blue.
In photon transport through matter, Rayleigh scattering is most relevant at relatively low photon energies. As photon energy increases, Compton scattering often becomes more important.
Summary relations
The most useful beginner level facts about Rayleigh scattering are collected below.
| Quantity | Rayleigh scattering result |
|---|---|
| Nature of process | Elastic scattering |
| Photon energy change | None |
| Main interacting charges | Bound electrons |
| Small-particle wavelength law | $I \propto 1/\lambda^4$ |
| Typical role in matter | Changes photon direction, contributes to attenuation from beam |
Core ideas of Rayleigh scattering:
$$E' = E$$
$$I \propto \frac{1}{\lambda^4} \quad \text{for small scatterers}$$
It is scattering by bound charges, and it mainly redirects photons without absorbing them.