Rayleigh scattering

Simply explained: the scattering of light by particles much smaller than its wavelength, such as air molecules or nanoparticles of a few nanometres. The light makes each particle a tiny oscillating dipole, which re-radiates; the scattered intensity grows as the sixth power of the size and the inverse fourth power of the wavelength.

For a sphere of diameter d and relative refractive index m: C_sca = (2π⁵/3) (d⁶/λ⁴) |(m²−1)/(m²+2)|², with λ the wavelength in the medium. The intensity at distance R and angle θ for unpolarized light is I = I₀ (1 + cos²θ)/(2R²) (2π/λ)⁴ |(m²−1)/(m²+2)|² (d/2)⁶.

During the day on Earth, sunlight crosses the atmosphere and its air molecules Rayleigh-scatter it, as 1/λ⁴: blue light at 450 nm is scattered about (700/450)⁴ ≈ 6 times more than red at 700 nm. Sunlight scattered out of its path towards us from all over the sky is therefore rich in blue.

Violet is scattered even more, but sunlight contains less violet than blue, ozone absorbs some of it, and our eyes are much less sensitive to violet. The mixture of scattered wavelengths is perceived as sky blue.

The blue is not in the sunlight itself but in the light scattered by air molecules. In space there is (almost) no air to scatter sunlight towards you, so you see only the direct sunlight and black around it. The Moon, with no atmosphere, has a black sky even in daylight.

At sunset the light crosses far more air to reach you. Rayleigh scattering removes the blue along the way, so what arrives directly from the sun is red and orange; dust and aerosols strengthen the effect.

Cloud droplets are 10–20 µm across, much larger than the wavelength, so (even in white, fluffy cumulus) they are far outside the Rayleigh regime: in the Mie / geometric-optics regime all visible wavelengths scatter almost equally, and the scattered light stays white. Thick clouds look grey because little light gets through.

Bulk water transmits light straight through. A cloud breaks the same water into billions of droplets about 10–20 µm across, each of which scatters light strongly and almost equally at all colours (Mie scattering); after many scatterings the light leaves in all directions, white. Thick clouds look grey only because less light gets through them.

Rayleigh scattering is the limit for particles much smaller than the wavelength (size parameter x = πd n_m/λ ≪ 1): 1/λ⁴, symmetric forward and backward. Mie theory is exact for spheres of any size and includes Rayleigh scattering as its small-size limit; for larger particles the scattering becomes forward-peaked and nearly colour-neutral, and metals and high-index particles show resonances.

When the particle is no longer much smaller than the wavelength inside it: roughly for x above 0.3, or when |m|x approaches 1. A 100 nm polystyrene bead in green light (x ≈ 0.76) already deviates noticeably; the Mie calculator gives the exact answer at any size.

The blue sky, red sunsets, the bluish tint of distant mountains and of smoke from small fires, the blue of some bird feathers and eyes (structural), and the light lost in optical fibers. Rayleigh scattering spectroscopy (Rayleigh and Raman lidar, light-scattering photometers) uses it to measure gas density and small particles.

Rayleigh scattering is elastic: the scattered light keeps its wavelength. Raman scattering (the Raman effect) is inelastic: a tiny fraction of the light (about one photon in 10⁶–10⁸) exchanges energy with molecular vibrations and comes out at shifted wavelengths, which identify the molecules.

A Rayleigh scatterer radiates like a dipole: light scattered at 90° to the sun's direction is almost fully linearly polarized (degree of polarization sin²θ / (1 + cos²θ) for unpolarized incident light). Bees and polarizing sunglasses use this.

The scattering of a light beam by colloidal particles in a liquid or gas, which makes the beam visible from the side, as in a sunbeam in a dusty room or a laser through a colloid. Small colloids scatter in the Rayleigh regime (bluish scattered light), larger ones in the Mie regime (whiter).