Rayleigh Scattering - Intensity of Light from molecules


Rayleigh scattering (pronounced /ˈreɪli/ RAY-lee), named after the British physicist Lord Rayleigh (John William Strutt), is the (dominantly) elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation. The Rayleigh scattering does not change the state of material hence it is 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” (i.e. with a refractive index close to 1). On the other hand, Anomalous Diffraction Theory applies to optically soft but larger particles.

Small size parameter approximation

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 (x ≪ 1, with a particle size < 1 /10 wavelength) 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 the formula shown in : “Rayleigh Scattering – Intensity of Light”

From molecules

The Intensity of Light expression 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 the formula shown here.

Related formulas


Iintensity of light scattered by a single particle (W/cm2)
I0intensity (W/cm2)
αmolecular polarizability (cm3)
λwavelength of the light (cm)
Rdistance to the particle (cm)
θscattering angle (deg)