---
title: "Rayleigh scattering"
chunk: 2/2
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"
---

  
    
      
        
          I
          
            s
          
        
        =
        
          I
          
            0
          
        
        
          
            
              
                π
                
                  2
                
              
              
                α
                
                  2
                
              
            
            
              
                
                  
                    ε
                    
                      0
                    
                  
                
                
                  2
                
              
              
                λ
                
                  4
                
              
              
                R
                
                  2
                
              
            
          
        
        
          
            
              1
              +
              
                cos
                
                  2
                
              
              ⁡
              θ
            
            2
          
        
        .
      
    
    {\displaystyle 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 
  
    
      
        ϵ
      
    
    {\displaystyle \epsilon }
  
 of a certain region of volume 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is different from the average dielectric constant of the medium 
  
    
      
        
          
            
              ϵ
              ¯
            
          
        
      
    
    {\displaystyle {\bar {\epsilon }}}
  
, then any incident light will be scattered according to the following equation

  
    
      
        I
        =
        
          I
          
            0
          
        
        
          
            
              
                π
                
                  2
                
              
              
                V
                
                  2
                
              
              
                σ
                
                  ϵ
                
                
                  2
                
              
            
            
              2
              
                λ
                
                  4
                
              
              
                R
                
                  2
                
              
            
          
        
        
          
            (
            
              1
              +
              
                cos
                
                  2
                
              
              ⁡
              θ
            
            )
          
        
      
    
    {\displaystyle I=I_{0}{\frac {\pi ^{2}V^{2}\sigma _{\epsilon }^{2}}{2\lambda ^{4}R^{2}}}{\left(1+\cos ^{2}\theta \right)}}
  
where 
  
    
      
        
          σ
          
            ϵ
          
          
            2
          
        
      
    
    {\displaystyle \sigma _{\epsilon }^{2}}
  
 represents the variance of the fluctuation in the dielectric constant 
  
    
      
        ϵ
      
    
    {\displaystyle \epsilon }
  
.

== 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:

  
    
      
        
          α
          
            scat
          
        
        =
        
          
            
              8
              
                π
                
                  3
                
              
            
            
              3
              
                λ
                
                  4
                
              
            
          
        
        
          n
          
            8
          
        
        
          p
          
            2
          
        
        k
        
          T
          
            f
          
        
        β
      
    
    {\displaystyle \alpha _{\text{scat}}={\frac {8\pi ^{3}}{3\lambda ^{4}}}n^{8}p^{2}kT_{\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 ==

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 ==
Rayleigh sky model – Polarization pattern of the daytime sky
Rician fading – Radio signal statistical model
Optical phenomena – Observable events that result from the interaction of light and matterPages displaying short descriptions of redirect targets
Dynamic light scattering – Technique for determining size distribution of particles
Raman scattering – Inelastic scattering of photons by matter
Rayleigh–Gans approximation
Tyndall effect – Scattering of light by tiny particles in a colloidal suspension
Critical opalescence – Increase in photonic scattering during a phase transition
HRS Computing – scientific simulation software
Marian Smoluchowski – Polish physicist (1872–1917)
Rayleigh criterion – Ability of any image-forming device to distinguish small details of an object
Aerial perspective – Atmospheric effects on the appearance of a distant object
Parametric process – Interacting phenomenon between light and matter
Bragg's law – Physical law regarding scattering angles of radiation through a medium

== Works ==
Strutt, J.W (1871). "XV. On the light from the sky, its polarization and colour". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (271): 107–120. doi:10.1080/14786447108640452.
Strutt, J.W (1871). "XXXVI. On the light from the sky, its polarization and colour". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (273): 274–279. doi:10.1080/14786447108640479.
Strutt, J.W (1871). "LVIII. On the scattering of light by small particles". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 41 (275): 447–454. doi:10.1080/14786447108640507.
Rayleigh, Lord (1881). "X. On the electromagnetic theory of light". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 12 (73): 81–101. doi:10.1080/14786448108627074.
Rayleigh, Lord (1899). "XXXIV. On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 47 (287): 375–384. doi:10.1080/14786449908621276.

== References ==

== Further reading ==

== External links ==

HyperPhysics description of Rayleigh scattering
Full physical explanation of sky color, in simple terms