Editing Rayleigh scattering

{{Short description|Light scattering by small particles}}
{{About|the optical phenomenon|the magnetic phenomenon|Rayleigh law|the stochastic distribution|Rayleigh distribution|the wireless communication effect|Rayleigh fading}}
[[File:Leehasacamera - Sunset over the clouds (by).jpg|thumb|300x300px|Rayleigh scattering causes the blue color of the daytime sky and the reddening of the Sun at sunset.]]
'''Rayleigh scattering''' ({{IPAc-en|ˈ|r|eɪ|l|i}} {{respell|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 relation|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).<ref>Lord Rayleigh (John Strutt) refined his theory of scattering in a series of papers; see [[Rayleigh scattering#Works|Works]].</ref> 

[[File: Monochrome Rainbow.jpg|thumb|300x300px|Due to Rayleigh scattering, red and orange colors are more visible during sunset because the blue and violet light has been scattered out of the direct path. Due to removal of such colors, these colors are scattered by [[Atmospheric_optics#Sky coloration|dramatically colored skies]] and [[Monochrome rainbow|monochromatic rainbows]].]]
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 [[gas]]es.

Rayleigh scattering of [[sunlight]] in [[Earth's atmosphere]] causes [[diffuse sky radiation]], which is the reason for the blue color of the [[daytime]] and [[twilight]] [[sky]], as well as the [[golden hour (photography)|yellowish]] to reddish hue of the low [[Sun]]. Sunlight is also subject to [[Raman scattering]], which changes the rotational state of the molecules and gives rise to [[polarization (waves)|polarization]] effects.<ref>{{cite journal|doi=10.1364/AO.20.000533|pmid=20309152|title=Rayleigh scattering|journal=Applied Optics|volume=20|issue=4|pages=533–5|year=1981|last1=Young|first1=Andrew T|bibcode=1981ApOpt..20..533Y}}</ref>

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.<ref>{{Cite journal |last1=Tyndall |first1=John |title=On the blue colour of the sky, the polarization of skylight, and on the polarization of light by cloudy matter generally |journal=Proceedings of the Royal Society of London |date=1869 |volume=17 |pages=223–233 |doi=10.1098/rspl.1868.0033|doi-access=free }}</ref> He conjectured that a similar scattering of sunlight gave the sky its [[Diffuse sky radiation|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 effect|Tyndall's effect]] in water droplets in terms of the tiny particulates' volumes and [[Refractive index|refractive indices]].<ref>{{cite journal |last1=Strutt |first1=Hon. J.W. |title=On the light from the sky, its polarization and colour |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1871 |volume=41 |issue=271 |pages=107–120 |doi=10.1080/14786447108640452}}</ref><ref>{{cite journal |last1=Strutt |first1=Hon. J.W. |title=On the light from the sky, its polarization and colour |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1871 |volume=41 |issue=273 |pages=274–279 |doi=10.1080/14786447108640479}}</ref><ref>{{cite journal |last1=Strutt |first1=Hon. J.W. |title=On the scattering of light by small particles |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1871 |volume=41 |issue=275 |pages=447–454 |doi=10.1080/14786447108640507}}</ref> In 1881, with the benefit of [[James Clerk Maxwell]]'s 1865 [[History of Maxwell's equations|proof of the electromagnetic nature of light]], he showed that his equations followed from [[electromagnetism]].<ref>{{cite journal |last1=Rayleigh |first1=Lord |title=On the electromagnetic theory of light |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1881 |volume=12 |issue=73 |pages=81–101 |doi=10.1080/14786448108627074|url=https://zenodo.org/record/1431155 }}</ref> In 1899, he showed that they applied to individual molecules, with terms containing particulate volumes and refractive indices replaced with terms for molecular [[polarizability]].<ref>{{cite journal |last1=Rayleigh |first1=Lord |title=On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1899 |volume=47 |issue=287 |pages=375–384 |doi=10.1080/14786449908621276|url=https://zenodo.org/record/1431249 }}</ref>

==Small size parameter approximation==
The size of a scattering particle is often parameterized by the ratio

<math display="block"> x = \frac{2 \pi r} {\lambda}</math>

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 (waves)|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<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html Blue Sky and Rayleigh Scattering]. Hyperphysics.phy-astr.gsu.edu. Retrieved on 2018-08-06.</ref>) 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 [[coherence (physics)|incoherent]] and the resulting [[intensity (physics)|intensity]] is just the sum of the squares of the [[amplitude]]s from each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size.<ref name="Cornell">{{Cite web |last=Rana |first=Farhan  |title=Electromagnetic Scattering |url=https://courses.cit.cornell.edu/ece303/Lectures/lecture34.pdf |access-date=2 April 2014 |website=ECE303 Electromagnetic Fields and Waves}}</ref><ref>{{cite journal|last=Barnett|first=C.E.|title=Some application of wavelength turbidimetry in the infrared|journal=J. Phys. Chem.|year=1942|volume=46|issue=1|pages=69–75|doi=10.1021/j150415a009}}</ref> The wavelength dependence is characteristic of [[Dipole radiation|dipole scattering]]<ref name=Cornell/> 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 ''I''<sub>0</sub> is given by<ref>Seinfeld, John H. and Pandis, Spyros N. (2006) ''Atmospheric Chemistry and Physics, 2nd Edition'', John Wiley and Sons, New Jersey, Chapter 15.1.1, {{ISBN|0471720186}}</ref>
<math display="block"> 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</math>
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:<ref>{{cite journal|last=Cox|first=A.J.|title=An experiment to measure Mie and Rayleigh total scattering cross sections|journal=American Journal of Physics|year=2002|volume=70|issue=6|page=620|doi=10.1119/1.1466815|bibcode = 2002AmJPh..70..620C |s2cid=16699491}}</ref>
<math display="block"> \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 .</math>
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%.<ref name=SneepUbacks/>

The fraction of light scattered by scattering particles over the unit travel length (e.g., meter) is the number of particles per unit volume ''N'' times the cross-section. For example, air has a refractive index of 1.0002793 at atmospheric pressure, where there are about {{val|2|e=25}} molecules per cubic meter, and therefore the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of {{val|5.1|e=-31|u=m<sup>2</sup>}} at a wavelength of 532&nbsp;nm (green light).<ref name=SneepUbacks>{{cite journal | last1 = Sneep | first1 = Maarten | last2 = Ubachs | first2 = Wim | year = 2005 | title = Direct measurement of the Rayleigh scattering cross section in various gases | doi = 10.1016/j.jqsrt.2004.07.025 | journal = Journal of Quantitative Spectroscopy and Radiative Transfer | volume = 92 | issue = 3| pages = 293–310 |bibcode = 2005JQSRT..92..293S }}</ref> This means that  about a fraction 10<sup>−5</sup> of the light will be scattered for every meter of travel.

The strong wavelength dependence of the scattering (~''λ''<sup>−4</sup>) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths.

==From molecules==
[[File:Rayleigh sunlight scattering.svg|thumb|Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light]]
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 [[Centimetre–gram–second system of units|CGS-units]] by<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html#c2 Rayleigh scattering]. Hyperphysics.phy-astr.gsu.edu. Retrieved on 2018-08-06.</ref>
<math display="block">I_s = I_0 \frac{8\pi^4\alpha^2}{\lambda^4 R^2}(1+\cos^2\theta)</math>
and in [[International System of Units|SI-units]] by
<math display="block">I_s = I_0 \frac{\pi^2\alpha^2}{{\varepsilon_0}^2 \lambda^4 R^2}\frac{1+\cos^2(\theta)}{2} .</math>

== Effect of fluctuations ==
When the [[dielectric constant]] <math>\epsilon</math> of a certain region of volume <math>V</math> is different from the average dielectric constant of the medium <math>\bar{\epsilon}</math>, then any incident light will be scattered according to the following equation<ref>{{Cite book|title=Statistical mechanics|last=McQuarrie, Donald A. (Donald Allan)|date=2000|publisher=University Science Books|isbn=1891389157|location=Sausalito, Calif.|pages=[https://archive.org/details/statisticalmecha00mcqu_0/page/62 62]|oclc=43370175|url=https://archive.org/details/statisticalmecha00mcqu_0/page/62}}</ref>

<math display="block">I=I_0\frac{\pi^2V^2\sigma_\epsilon^2}{2\lambda^4R^2} {\left (1+\cos^2\theta\right )}</math>where <math>\sigma_\epsilon^2</math> represents the [[variance]] of the fluctuation in the dielectric constant <math>\epsilon</math>.

== Cause of the blue color of the sky==
{{Main|Diffuse sky radiation}}

[[File:CircularPolarizer.jpg|thumb|right|Scattered blue light is [[Polarization (waves)|polarized]]. The picture on the right is shot through a [[Polarizing filter (photography)|polarizing filter]]: the [[polarizer]] transmits light that is [[Linear polarization|linearly polarized]] in a specific direction.]]
The blue color of the sky is a consequence of three factors:<ref name=SmithAJP2005>{{Cite journal |last=Smith |first=Glenn S. |date=2005-07-01 |title=Human color vision and the unsaturated blue color of the daytime sky |url=https://pubs.aip.org/ajp/article/73/7/590/1056162/Human-color-vision-and-the-unsaturated-blue-color |journal=American Journal of Physics |language=en |volume=73 |issue=7 |pages=590–597 |doi=10.1119/1.1858479 |bibcode=2005AmJPh..73..590S |issn=0002-9505|url-access=subscription }}</ref> 
*the [[blackbody radiation | 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''<sup>−4</sup>) 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.<ref name=SmithAJP2005/>

Some of the scattering can also be from sulfate particles. For years after large [[Plinian eruption]]s, 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.<ref name="zerefos">{{Citation
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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]]).<ref>{{Citation |last=Choudhury |first=Asim Kumar Roy |title=Unusual visual phenomena and colour blindness |date=2014 |url=https://linkinghub.elsevier.com/retrieve/pii/B978085709229850005X |work=Principles of Colour and Appearance Measurement |pages=185–220 |publisher=Elsevier |language=en |doi=10.1533/9780857099242.185 |isbn=978-0-85709-229-8 |access-date=2022-03-29|url-access=subscription }}</ref>

==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.<ref>{{cite journal |last1=Mahajan |first1=Shivam |last2=Pica Ciamarra |first2=Massimo |title=Quasi-localized vibrational modes, boson peak and sound attenuation in model mass-spring networks |journal=SciPost Physics |year=2023 |volume=15 |issue=2 |page=069 |doi=10.21468/SciPostPhys.15.2.069 |arxiv=2211.01137 |doi-access=free |bibcode=2023ScPP...15...69M }}</ref> This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~''λ''<sup>−2</sup> 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:<ref>Rajagopal, K. (2008) ''Textbook on Engineering Physics'', PHI, New Delhi, part I, Ch. 3, {{ISBN|8120336658}}</ref>
<math display="block">\alpha_\text{scat} = \frac{8 \pi^3}{3 \lambda^4} n^8 p^2 k T_\text{f} \beta</math>

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''<sub>f</sub> is a ''fictive temperature'', representing the temperature at which the density fluctuations are "frozen" in the material.

==In porous materials==
[[File:Why is the sky blue.jpg|thumb|Rayleigh scattering in [[Opalescence|opalescent]] glass: it appears blue from the side, but orange light shines through.<ref>[http://www.webexhibits.org/causesofcolor/14B.html Blue & red | Causes of Color]. Webexhibits.org. Retrieved on 2018-08-06.</ref>]]
Rayleigh-type ''λ''<sup>−4</sup> scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials.<ref name = Svensson>{{cite journal|doi=10.1063/1.3292210|title=Laser spectroscopy of gas confined in nanoporous materials|journal=Applied Physics Letters|volume=96|issue=2|pages=021107|year=2010|last1=Svensson|first1=Tomas|last2=Shen|first2=Zhijian|url=http://lup.lub.lu.se/search/ws/files/2070270/2370765.pdf|bibcode=2010ApPhL..96b1107S|arxiv=0907.5092|s2cid=53705149}}</ref> 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''<sup>−4</sup>-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70&nbsp;nm) obtained by [[sintering]] monodispersive alumina powder.

==See also==
* {{annotated link|Rayleigh sky model}}
* {{annotated link|Rician fading}}
* {{annotated link|Optical phenomena}}
* {{annotated link|Dynamic light scattering}}
* {{annotated link|Raman scattering}}
* {{annotated link|Rayleigh–Gans approximation}}
* {{annotated link|Tyndall effect}}
* {{annotated link|Critical opalescence}}
* [[HRS Computing]] – scientific simulation software
* {{annotated link|Marian Smoluchowski}}
* {{annotated link|Angular resolution#The Rayleigh criterion|Rayleigh criterion}}
* {{annotated link|Aerial perspective}}
* {{annotated link|Parametric process (optics)|Parametric process}}
* {{annotated link|Bragg's law}}

==Works==

* {{cite journal|doi=10.1080/14786447108640452|title=XV. On the light from the sky, its polarization and colour|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=41|issue=271|pages=107–120|year=1871|last1=Strutt|first1=J.W}}
* {{cite journal|doi=10.1080/14786447108640479|title=XXXVI. On the light from the sky, its polarization and colour|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=41|issue=273|pages=274–279|year=1871|last1=Strutt|first1=J.W}}
* {{cite journal|doi=10.1080/14786447108640507|title=LVIII. On the scattering of light by small particles|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=41|issue=275|pages=447–454|year=1871|last1=Strutt|first1=J.W}}
* {{cite journal|doi=10.1080/14786448108627074|title=X. On the electromagnetic theory of light|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=12|issue=73|pages=81–101|year=1881|last1=Rayleigh|first1=Lord|url=https://zenodo.org/record/1431155}}
* {{cite journal|doi=10.1080/14786449908621276|title=XXXIV. On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=47|issue=287|pages=375–384|year=1899|last1=Rayleigh|first1=Lord|url=https://zenodo.org/record/1431249}}

==References==
{{Reflist}}

{{Commons category|Atmospheric Rayleigh scattering}}

==Further reading==
{{Refbegin}}
*C.F. Bohren, D. Huffman, ''Absorption and scattering of light by small particles'', John Wiley, New York 1983. Contains a good description of the asymptotic behavior of Mie theory for small size parameter (Rayleigh approximation).
* {{cite book | last = Ditchburn | first =  R.W. | year = 1963 | title = Light |url=https://archive.org/details/light0000ditc | url-access = registration | edition = 2nd | pages = [https://archive.org/details/light0000ditc/page/582 582–585] | publisher = Blackie & Sons | location = London | isbn = 978-0-12-218101-6 }}
* {{cite journal | last = Chakraborti | first = Sayan |date=September 2007 | title = Verification of the Rayleigh scattering cross section | journal = [[American Journal of Physics]] | volume = 75 | issue = 9 | pages = 824–826 | doi = 10.1119/1.2752825 |arxiv = physics/0702101 |bibcode = 2007AmJPh..75..824C | s2cid = 119100295 }}
* {{cite book | last = Ahrens | first = C. Donald | year = 1994 | title = Meteorology Today: an introduction to weather, climate, and the environment | edition = 5th | pages = [https://archive.org/details/meteorologytoday00ahre/page/88 88–89] | publisher = West Publishing Company | location = St. Paul MN | isbn = 978-0-314-02779-5 |url=https://archive.org/details/meteorologytoday00ahre/page/88 }}
* {{cite journal | last1 = Lilienfeld | first1 = Pedro | year = 2004| title = A Blue Sky History | journal = Optics and Photonics News | volume = 15 | issue = 6| pages = 32–39 | doi = 10.1364/OPN.15.6.000032 }} Gives a brief history of theories of why the sky is blue leading up to Rayleigh's discovery, and a brief description of Rayleigh scattering.
{{Refend}}

==External links==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html#c2 HyperPhysics description of Rayleigh scattering]
* [http://math.ucr.edu/home/baez/physics/General/BlueSky/blue_sky.html Full physical explanation of sky color, in simple terms]

{{DEFAULTSORT:Rayleigh Scattering}}
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