Rayleigh scattering

Small size parameter approximation

The size of a scattering particle is often parameterized by the ratio

$${\displaystyle x=\frac{2\pi r}{\lambda }}$$

where r is the particle's radius, λ is the wavelength of the light and x is a dimensionless parameter that characterizes the particle's interaction with the incident radiation such that: 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 of 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. The wavelength dependence is characteristic of dipole scattering and the volume dependence will apply to any scattering mechanism. In detail, the intensity of light scattered by any one of the small spheres of radius r and refractive index n from a beam of unpolarized light of wavelength λ and intensity I₀ is given by $${\displaystyle I_s=I_0\frac{1+{\cos }^2\theta }{2R^2}{\left(\frac{2\pi }{\lambda }\right)}^4{\left(\frac{n^2-1}{n^2+2}\right)}^2{r}^6}$$ where R is the observer's distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh scattering cross-section of the particles in air: $${\displaystyle {\sigma }_{\text{s}}=\frac{8\pi }{3}{\left(\frac{2\pi }{\lambda }\right)}^4{\left(\frac{n^2-1}{n^2+2}\right)}^2{r}^6.}$$ Here n is the refractive index of the spheres that approximate the molecules of the gas; the index of the gas surrounding the spheres is neglected, an approximation that introduces an error of less than 0.05%.

The major constituent of the atmosphere, nitrogen, has Rayleigh cross section of 5.1×10⁻³¹ m² at a wavelength of 532 nm (green light). Over the length of one meter the fraction of light scattered can be approximated from the product of the cross-section and the particle density, that is number of particles per unit volume. For air at atmospheric pressure there are about 2×10²⁵ molecules per cubic meter, and the fraction scattered will be 10⁻⁵ for every meter of travel.

From molecules

The expression above 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 $${\displaystyle I_s=I_0\frac{8{\pi }^4{\alpha }^2}{{\lambda }^4{R}^2}(1+{\cos }^2\theta )}$$ and in SI-units by $${\displaystyle I_s=I_0\frac{{\pi }^2{\alpha }^2}{{{\epsilon }_0}^2{\lambda }^4{R}^2}\frac{1+{\cos }^2\theta }{2}.}$$

Effect of fluctuations

When the dielectric constant ${\displaystyle ϵ}$ of a certain region of volume ${\displaystyle V}$ is different from the average dielectric constant of the medium ${\displaystyle \overline{ϵ}}$, then any incident light will be scattered according to the following equation

$${\displaystyle I=I_0\frac{{\pi }^2{V}^2{\sigma }_{ϵ}^2}{2{\lambda }^4{R}^2}\left(1+{\cos }^2\theta \right)}$$where ${\displaystyle {\sigma }_{ϵ}^2}$ represents the variance of the fluctuation in the dielectric constant ${\displaystyle ϵ}$.

Cause of the blue color of the sky

The blue color of the sky is a consequence of three factors:

• the blackbody spectrum of sunlight coming into the Earth's atmosphere,
• Rayleigh scattering of that light off oxygen and nitrogen molecules, and
• the response of the human visual system.

The strong wavelength dependence of the Rayleigh scattering (~λ⁻⁴) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths. This results in the indirect blue and violet light coming from all regions of the sky. The human eye responds to this wavelength combination as if it were a combination of blue and white light.

Some of the scattering can also be from sulfate particles. For years after large Plinian eruptions, the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime.

In locations with little light pollution, the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the Moon. The moonlit sky is not perceived as blue, however, because at low light levels human vision comes mainly from rod cells that do not produce any color perception (Purkinje effect).

Of sound in amorphous solids

Rayleigh scattering is also an important mechanism of wave scattering in amorphous solids such as glass, and is responsible for acoustic wave damping and phonon damping in glasses and granular matter at low or not too high temperatures. This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~λ⁻² dependence on wavelength), which becomes increasingly more important as the temperature rises.

In amorphous solids – glasses – optical fibers

Rayleigh scattering is an important component of the scattering of optical signals in optical fibers. Silica fibers are glasses, disordered materials with microscopic variations of density and refractive index. These give rise to energy losses due to the scattered light, with the following coefficient: $${\displaystyle {\alpha }_{\text{scat}}=\frac{8{\pi }^3}{3{\lambda }^4}{n}^8{p}^{2}k{T}_{\text{f}}\beta }$$

where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant, and β is the isothermal compressibility. T_f is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material.

In porous materials

Rayleigh-type λ⁻⁴ scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials. The strong contrast in refractive index between pores and solid parts of sintered alumina results in very strong scattering, with light completely changing direction each five micrometers on average. The λ⁻⁴-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained by sintering monodispersive alumina powder.