Rayleigh Scattering - Small Size Parameter Approximation

Small Size Parameter Approximation

The size of a scattering particle is parameterized by the ratio x of its characteristic dimension r and wavelength λ:

Rayleigh scattering can be defined as scattering in the small size parameter regime x ≪ 1. Scattering from larger spherical particles is explained by the Mie theory for an arbitrary size parameter x. For small x the Mie theory reduces to the Rayleigh approximation.

The amount of Rayleigh scattering that occurs for a beam of light depends upon the size of the particles and the wavelength of the light. Specifically, the intensity of the scattered light varies as the sixth power of the particle size, and varies inversely with the fourth power of the wavelength.

The intensity I of light scattered by a single small particle from a beam of unpolarized light of wavelength λ and intensity I₀ is given by:

where R is the distance to the particle, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle. The Rayleigh scattering cross-section is given by

The Rayleigh scattering coefficient for a group of scattering particles is the number of particles per unit volume N times the cross-section. As with all wave effects, for incoherent scattering the scattered powers add arithmetically, while for coherent scattering, such as if the particles are very near each other, the fields add arithmetically and the sum must be squared to obtain the total scattered power.