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Rayleigh scattering

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Rayleigh scattering causes the blue color of the daytime sky and the reddening of the Sun at sunset.

Rayleigh scattering(/ˈrli/RAY-lee), named after the 19th-century British physicistLord Rayleigh(John William Strutt),[1]is the predominantlyelastic scatteringoflight,or otherelectromagnetic radiation,by particles with a size much smaller than thewavelengthof the radiation. For light frequencies well below theresonancefrequency of the scattering medium (normaldispersionregime), the amount of scattering isinversely proportionalto thefourth powerof the wavelength, e.g., a blue color is scattered much more than a red color as light propagates through air.

Rayleigh scattering results from the electricpolarizabilityof 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 radiatingdipolewhose 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 ingases.

Rayleigh scattering ofsunlightinEarth's atmospherecausesdiffuse sky radiation,which is the reason for the blue color of thedaytimeandtwilightsky,as well as theyellowishto reddish hue of the lowSun.Sunlight is also subject toRaman scattering,which changes the rotational state of the molecules and gives rise topolarizationeffects.[2]

Scattering by particles with a size comparable to, or larger than, the wavelength of the light is typically treated by theMie theory,thediscrete dipole approximationand 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 arefractive indexclose to 1).Anomalous diffraction theoryapplies to optically soft but larger particles.

History[edit]

In 1869, while attempting to determine whether any contaminants remained in the purified air he used for infrared experiments,John Tyndalldiscovered that bright light scattering off nanoscopic particulates was faintly blue-tinted.[3][4]He conjectured that a similar scattering of sunlight gave the sky itsblue 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 Rayleighpublished two papers on the color and polarization of skylight to quantifyTyndall's effectin water droplets in terms of the tiny particulates' volumes andrefractive indices.[5][6][7]In 1881, with the benefit ofJames Clerk Maxwell's 1865proof of the electromagnetic nature of light,he showed that his equations followed fromelectromagnetism.[8]In 1899, he showed that they applied to individual molecules, with terms containing particulate volumes and refractive indices replaced with terms for molecularpolarizability.[9]

Small size parameter approximation[edit]

The size of a scattering particle is often parameterized by the ratio

whereris the particle's radius,λis thewavelengthof the light andxis adimensionless parameterthat 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 ofMie scattering,interference effects develop throughphasevariations 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[10]) 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 isincoherentand the resultingintensityis just the sum of the squares of theamplitudesfrom each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size.[11][12]The wavelength dependence is characteristic ofdipole scattering[11]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 diameterdandrefractive indexnfrom a beam of unpolarized light of wavelengthλand intensityI0is given by[13] whereRis the distance to the particle andθis the scattering angle. Averaging this over all angles gives the Rayleighscattering cross-sectionof the particles in air:[14] Herenis 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%.[15]

The fraction of light scattered by scattering particles over the unit travel length (e.g., meter) is the number of particles per unit volumeNtimes the cross-section. For example, air has a refractive index of 1.0002793 at atmospheric pressure, where there are about2×1025molecules per cubic meter, and therefore the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of5.1×10−31m2at a wavelength of 532 nm (green light).[15]This means that about a fraction 10−5of the light will be scattered for every meter of travel.

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

From molecules[edit]

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 molecularpolarizabilityα,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 inCGS-unitsby[16] and inSI-unitsby .

Effect of fluctuations[edit]

When thedielectric constantof a certain region of volumeis different from the average dielectric constant of the medium,then any incident light will be scattered according to the following equation[17]

whererepresents thevarianceof the fluctuation in the dielectric constant.

Cause of the blue color of the sky[edit]

Scattered blue light ispolarized.The picture on the right is shot through apolarizing filter:thepolarizertransmits light that islinearly polarizedin a specific direction.

The blue color of the sky is a consequence of three factors:[18]

  • theblackbodyspectrum ofsunlightcoming 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.[18]

Some of the scattering can also be from sulfate particles. For years after largePlinian eruptions,the blue cast of the sky is notably brightened by the persistent sulfate load of thestratosphericgases. Some works of the artistJ. M. W. Turnermay owe their vivid red colours to the eruption ofMount Tamborain his lifetime.[19]

In locations with littlelight pollution,the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lowercolor temperaturedue 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 fromrod cellsthat do not produce any color perception (Purkinje effect).[20]

Of sound in amorphous solids[edit]

Rayleigh scattering is also an important mechanism of wave scattering inamorphous solidssuch as glass, and is responsible for acoustic wave damping and phonon damping in glasses and granular matter at low or not too high temperatures.[21]This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~λ−2dependence on wavelength), which becomes increasingly more important as the temperature rises.

In amorphous solids – glasses – optical fibers[edit]

Rayleigh scattering is an important component of the scattering of optical signals inoptical 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:[22]

wherenis the refraction index,pis the photoelastic coefficient of the glass,kis theBoltzmann constant,andβis the isothermal compressibility.Tfis afictive temperature,representing the temperature at which the density fluctuations are "frozen" in the material.

In porous materials[edit]

Rayleigh scattering inopalescentglass: it appears blue from the side, but orange light shines through.[23]

Rayleigh-typeλ−4scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials.[24]The strong contrast in refractive index between pores and solid parts of sinteredaluminaresults in very strong scattering, with light completely changing direction each five micrometers on average. Theλ−4-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained bysinteringmonodispersive alumina powder.

See also[edit]

Works[edit]

  • 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[edit]

  1. ^Lord Rayleigh (John Strutt) refined his theory of scattering in a series of papers; seeWorks.
  2. ^Young, Andrew T (1981). "Rayleigh scattering".Applied Optics.20(4): 533–5.Bibcode:1981ApOpt..20..533Y.doi:10.1364/AO.20.000533.PMID20309152.
  3. ^Tyndall, John (1869)."On the blue colour of the sky, the polarization of skylight, and on the polarization of light by cloudy matter generally".Proceedings of the Royal Society of London.17:223–233.doi:10.1098/rspl.1868.0033.
  4. ^Conocimiento, Ventana al (2018-08-01)."John Tyndall, the Man who Explained Why the Sky is Blue".OpenMind.Retrieved2019-03-31.
  5. ^Strutt, Hon. J.W. (1871). "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.
  6. ^Strutt, Hon. J.W. (1871). "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.
  7. ^Strutt, Hon. J.W. (1871). "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.
  8. ^Rayleigh, Lord (1881)."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.
  9. ^Rayleigh, Lord (1899)."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.
  10. ^Blue Sky and Rayleigh Scattering.Hyperphysics.phy-astr.gsu.edu. Retrieved on 2018-08-06.
  11. ^ab"Cornell lectures"(PDF).Retrieved2 April2014.
  12. ^Barnett, C.E. (1942). "Some application of wavelength turbidimetry in the infrared".J. Phys. Chem.46(1): 69–75.doi:10.1021/j150415a009.
  13. ^Seinfeld, John H. and Pandis, Spyros N. (2006)Atmospheric Chemistry and Physics, 2nd Edition,John Wiley and Sons, New Jersey, Chapter 15.1.1,ISBN0471720186
  14. ^Cox, A.J. (2002). "An experiment to measure Mie and Rayleigh total scattering cross sections".American Journal of Physics.70(6): 620.Bibcode:2002AmJPh..70..620C.doi:10.1119/1.1466815.S2CID16699491.
  15. ^abSneep, Maarten; Ubachs, Wim (2005). "Direct measurement of the Rayleigh scattering cross section in various gases".Journal of Quantitative Spectroscopy and Radiative Transfer.92(3): 293–310.Bibcode:2005JQSRT..92..293S.doi:10.1016/j.jqsrt.2004.07.025.
  16. ^Rayleigh scattering.Hyperphysics.phy-astr.gsu.edu. Retrieved on 2018-08-06.
  17. ^McQuarrie, Donald A. (Donald Allan) (2000).Statistical mechanics.Sausalito, Calif.: University Science Books. pp.62.ISBN1891389157.OCLC43370175.
  18. ^abSmith, Glenn S. (2005-07-01)."Human color vision and the unsaturated blue color of the daytime sky".American Journal of Physics.73(7): 590–597.doi:10.1119/1.1858479.ISSN0002-9505.
  19. ^Zerefos, C. S.; Gerogiannis, V. T.; Balis, D.; Zerefos, S. C.; Kazantzidis, A. (2007),"Atmospheric effects of volcanic eruptions as seen by famous artists and depicted in their paintings"(PDF),Atmospheric Chemistry and Physics,7(15): 4027–4042,Bibcode:2007ACP.....7.4027Z,doi:10.5194/acp-7-4027-2007
  20. ^Choudhury, Asim Kumar Roy (2014),"Unusual visual phenomena and colour blindness",Principles of Colour and Appearance Measurement,Elsevier, pp. 185–220,doi:10.1533/9780857099242.185,ISBN978-0-85709-229-8,retrieved2022-03-29
  21. ^Mahajan, Shivam; Pica Ciamarra, Massimo (2023)."Quasi-localized vibrational modes, boson peak and sound attenuation in model mass-spring networks".SciPost Physics.15(2).arXiv:2211.01137.doi:10.21468/SciPostPhys.15.2.069.
  22. ^Rajagopal, K. (2008)Textbook on Engineering Physics,PHI, New Delhi, part I, Ch. 3,ISBN8120336658
  23. ^Blue & red | Causes of Color.Webexhibits.org. Retrieved on 2018-08-06.
  24. ^Svensson, Tomas; Shen, Zhijian (2010)."Laser spectroscopy of gas confined in nanoporous materials"(PDF).Applied Physics Letters.96(2): 021107.arXiv:0907.5092.Bibcode:2010ApPhL..96b1107S.doi:10.1063/1.3292210.S2CID53705149.

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