Rayleigh scattering

In 1869, while attempting to determine whether any contaminants remained in the purified air he used for infrared experiments, John Tyndall discovered that bright light scattering off nanoscopic particulates was faintly blue-tinted. He conjectured that a similar scattering of sunlight gave the sky its blue hue, but he could not explain the preference for blue light, nor could atmospheric dust explain the intensity of the sky's color. In 1881, with the benefit of James Clerk Maxwell's 1865 proof of the electromagnetic nature of light, he showed that his equations followed from electromagnetism. In 1899, he showed that they applied to individual molecules, with terms containing particulate volumes and refractive indices replaced with terms for molecular polarizability. It was this paper that established the basic scientific model for the color of the sky. ==Small size parameter approximation==

The size of a scattering particle is often parameterized by the ratio 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 I_s = I_0 \frac{ 1+\cos^2 \theta }{2 R^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: \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%. 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 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 I_s = I_0 \frac{8\pi^4\alpha^2}{\lambda^4 R^2}(1+\cos^2\theta) and in SI-units by I_s = I_0 \frac{\pi^2\alpha^2}{{\varepsilon_0}^2 \lambda^4 R^2}\frac{1+\cos^2\theta}{2} . == Effect of fluctuations ==

When the dielectric constant \epsilon of a certain region of volume V is different from the average dielectric constant of the medium \bar{\epsilon}, then any incident light will be scattered according to the following equation I=I_0\frac{\pi^2V^2\sigma_\epsilon^2}{2\lambda^4R^2} {\left (1+\cos^2\theta\right )}where \sigma_\epsilon^2 represents the variance of the fluctuation in the dielectric constant \epsilon. == Cause of the blue color of the sky==

. The picture on the right is shot through a polarizing filter: the polarizer transmits light that is linearly polarized in a specific direction. 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. 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: \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. Tf is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material. ==In porous materials==

glass: it appears blue from the side, but orange light shines through. 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. ==See also==