---
title: "Rayleigh scattering"
chunk: 1/2
source: "https://en.wikipedia.org/wiki/Rayleigh_scattering"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:50:50.823381+00:00"
instance: "kb-cron"
---

Rayleigh scattering ( RAY-lee) is the scattering or deflection of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering medium (normal dispersion regime), the amount of scattering is inversely proportional to the fourth power of the wavelength (e.g., a blue color is scattered much more than a red color as light propagates through air). The phenomenon is named after the 19th-century British physicist Lord Rayleigh (John William Strutt). 

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. 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 of sunlight in Earth's atmosphere causes diffuse sky radiation. Since blue light wavelengths scatter more, the diffuse sky seen in daytime is blue. At twilight the sunlight on the horizon is missing the scattered blue light wavelengths giving a yellowish to reddish hue to the low Sun.
Scattering by particles with a size comparable to, or larger than, the wavelength of the 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). Anomalous diffraction theory applies to optically soft but larger particles.

== History ==
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 1871, Lord Rayleigh published two papers on the color and polarization of skylight to quantify Tyndall's effect in water droplets in terms of the tiny particulates' volumes and refractive indices. 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
        =
        
          
            
              2
              π
              r
            
            λ
          
        
      
    
    {\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

  
    
      
        
          I
          
            s
          
        
        =
        
          I
          
            0
          
        
        
          
            
              1
              +
              
                cos
                
                  2
                
              
              ⁡
              θ
            
            
              2
              
                R
                
                  2
                
              
            
          
        
        
          
            (
            
              
                
                  2
                  π
                
                λ
              
            
            )
          
          
            4
          
        
        
          
            (
            
              
                
                  
                    n
                    
                      2
                    
                  
                  −
                  1
                
                
                  
                    n
                    
                      2
                    
                  
                  +
                  2
                
              
            
            )
          
          
            2
          
        
        
          r
          
            6
          
        
      
    
    {\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:

  
    
      
        
          σ
          
            s
          
        
        =
        
          
            
              8
              π
            
            3
          
        
        
          
            (
            
              
                
                  2
                  π
                
                λ
              
            
            )
          
          
            4
          
        
        
          
            (
            
              
                
                  
                    n
                    
                      2
                    
                  
                  −
                  1
                
                
                  
                    n
                    
                      2
                    
                  
                  +
                  2
                
              
            
            )
          
          
            2
          
        
        
          r
          
            6
          
        
        .
      
    
    {\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

  
    
      
        
          I
          
            s
          
        
        =
        
          I
          
            0
          
        
        
          
            
              8
              
                π
                
                  4
                
              
              
                α
                
                  2
                
              
            
            
              
                λ
                
                  4
                
              
              
                R
                
                  2
                
              
            
          
        
        (
        1
        +
        
          cos
          
            2
          
        
        ⁡
        θ
        )
      
    
    {\displaystyle I_{s}=I_{0}{\frac {8\pi ^{4}\alpha ^{2}}{\lambda ^{4}R^{2}}}(1+\cos ^{2}\theta )}
  

and in SI-units by