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Scattering

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For other uses, seeScattering (disambiguation).

In physics,scatteringis a wide range of physical processes where moving particles or radiation of some form, such aslightorsound,are forced to deviate from a straighttrajectoryby localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by thelaw of reflection.Reflections of radiation that undergo scattering are often calleddiffuse reflectionsand unscattered reflections are calledspecular(mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far asIsaac Newtonin the 17th century[1]). As more "ray" -like phenomena were discovered, the idea of scattering was extended to them, so thatWilliam Herschelcould refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800.[2]John Tyndall,a pioneer in light scattering research, noted the connection between light scattering and acoustic scattering in the 1870s.[3]Near the end of the 19th century, the scattering ofcathode rays(electron beams)[4]and X-rays[5]was observed and discussed. With the discovery of subatomic particles (e.g.Ernest Rutherfordin 1911[6]) and the development of quantum theory in the 20th century, the sense of the term became broader as it was recognized that the same mathematical frameworks used in light scattering could be applied to many other phenomena.

Wine glassinLCD projectorslight beam makes the beam scatter.

Scattering can refer to the consequences ofparticle-particle collisionsbetween molecules, atoms,electrons,photons and other particles. Examples include:cosmic rayscattering in the Earth's upper atmosphere; particle collisions insideparticle accelerators;electron scattering by gas atoms in fluorescent lamps; andneutron scatteringinsidenuclear reactors.[7]

The types of non-uniformities which can cause scattering, sometimes known asscatterersorscattering centers,are too numerous to list, but a small sample includesparticles,bubbles,droplets,densityfluctuations influids,crystallitesinpolycrystallinesolids, defects inmonocrystallinesolids,surface roughness,cellsin organisms, and textilefibersin clothing. The effects of such features on the path of almost any type of propagating wave or moving particle can be described in the framework ofscattering theory.

Some areas where scattering and scattering theory are significant include radar sensing,medical ultrasound,semiconductor waferinspection,polymerizationprocess monitoring, acoustic tiling, free-space communications andcomputer-generated imagery.[8]Particle-particle scattering theory is important in areas such asparticle physics,atomic, molecular, and optical physics,nuclear physicsandastrophysics.Inparticle physicsthe quantum interaction and scattering of fundamental particles is described by the Scattering Matrix orS-Matrix,introduced and developed byJohn Archibald WheelerandWerner Heisenberg.[9]

Scattering is quantified using many different concepts, includingscattering cross section(σ),attenuation coefficients,thebidirectional scattering distribution function(BSDF),S-matrices,andmean free path.

Single and multiple scattering

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Zodiacal lightis a faint, diffuse glow visible in thenight sky.The phenomenon stems from thescatteringofsunlightbyinterplanetary dustspread throughoutthe planeof theSolar System.[10]

When radiation is only scattered by one localized scattering center, this is calledsingle scattering.It is more common that scattering centers are grouped together; in such cases, radiation may scatter many times, in what is known asmultiple scattering.[11]The main difference between the effects of single and multiple scattering is that single scattering can usually be treated as a random phenomenon, whereas multiple scattering, somewhat counterintuitively, can be modeled as a more deterministic process because the combined results of a large number of scattering events tend to average out. Multiple scattering can thus often be modeled well withdiffusion theory.[12]

Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome, which tends to depend strongly on the exact incoming trajectory, appears random to an observer. This type of scattering would be exemplified by an electron being fired at an atomic nucleus. In this case, the atom's exact position relative to the path of the electron is unknown and would be unmeasurable, so the exact trajectory of the electron after the collision cannot be predicted. Single scattering is therefore often described by probability distributions.

With multiple scattering, the randomness of the interaction tends to be averaged out by a large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by alight beampassing through thickfog.Multiple scattering is highly analogous todiffusion,and the termsmultiple scatteringanddiffusionare interchangeable in many contexts. Optical elements designed to produce multiple scattering are thus known asdiffusers.[13]Coherent backscattering,an enhancement ofbackscatteringthat occurs when coherent radiation is multiply scattered by a random medium, is usually attributed toweak localization.

Not all single scattering is random, however. A well-controlled laser beam can be exactly positioned to scatter off a microscopic particle with a deterministic outcome, for instance. Such situations are encountered inradarscattering as well, where the targets tend to be macroscopic objects such as people or aircraft.

Similarly, multiple scattering can sometimes have somewhat random outcomes, particularly with coherent radiation. The random fluctuations in the multiply scattered intensity of coherent radiation are calledspeckles.Speckle also occurs if multiple parts of a coherent wave scatter from different centers. In certain rare circumstances, multiple scattering may only involve a small number of interactions such that the randomness is not completely averaged out. These systems are considered to be some of the most difficult to model accurately.

The description of scattering and the distinction between single and multiple scattering are tightly related towave–particle duality.

Theory

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Scattering theory is a framework for studying and understanding the scattering ofwavesandparticles.Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance (sunlight) scattered byrain dropsto form arainbow.Scattering also includes the interaction ofbilliard ballson a table, theRutherford scattering(or angle change) ofalpha particlesbygoldnuclei,the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and theinelastic scatteringof a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions ofpartial differential equations,propagating freely "in the distant past", come together and interact with one another or with aboundary condition,and then propagate away "to the distant future".

Thedirect scattering problemis the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. Theinverse scattering problemis the problem of determining the characteristics of an object (e.g., its shape, internal constitution) from measurement data of radiation or particles scattered from the object.

Attenuation due to scattering

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Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units

When the target is a set of many scattering centers whose relative position varies unpredictably, it is customary to think of a range equation whose arguments take different forms in different application areas. In the simplest case consider an interaction that removes particles from the "unscattered beam" at a uniform rate that is proportional to the incident number of particles per unit area per unit time (), i.e. that

whereQis an interaction coefficient andxis the distance traveled in the target.

The above ordinary first-orderdifferential equationhas solutions of the form:

whereIois the initial flux, path length Δx ≡x−xo,the second equality defines an interactionmean free pathλ, the third uses the number of targets per unit volume η to define an areacross-sectionσ, and the last uses the target mass density ρ to define a density mean free path τ. Hence one converts between these quantities viaQ= 1/λ=ησ=ρ/τ,as shown in the figure at left.

In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm−1) is variously calledopacity,absorption coefficient,andattenuation coefficient.In nuclear physics, area cross-sections (e.g. σ inbarnsor units of 10−24cm2), density mean free path (e.g. τ in grams/cm2), and its reciprocal themass attenuation coefficient(e.g. in cm2/gram) orarea per nucleonare all popular, while in electron microscopy theinelastic mean free path[14](e.g. λ in nanometers) is often discussed[15]instead.

Elastic and inelastic scattering

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The term "elastic scattering" implies that the internal states of the scattering particles do not change, and hence they emerge unchanged from the scattering process. In inelastic scattering, by contrast, the particles' internal state is changed, which may amount to exciting some of the electrons of a scattering atom, or the complete annihilation of a scattering particle and the creation of entirely new particles.

The example of scattering inquantum chemistryis particularly instructive, as the theory is reasonably complex while still having a good foundation on which to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being thebound statesolutions of some differential equation. Thus, for example, thehydrogen atomcorresponds to a solution to theSchrödinger equationwith a negative inverse-power (i.e., attractive Coulombic)central potential.The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or evenionized,representing an inelastic scattering process.

The term "deep inelastic scattering"refers to a special kind of scattering experiment in particle physics.

Mathematical framework

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Inmathematics,scattering theory deals with a more abstract formulation of the same set of concepts. For example, if adifferential equationis known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role oftime.One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".

Solutions to differential equations are often posed onmanifolds.Frequently, the means to the solution requires the study of thespectrumof anoperatoron the manifold. As a result, the solutions often have a spectrum that can be identified with aHilbert space,and scattering is described by a certain map, theS matrix,on Hilbert spaces. Solutions with adiscrete spectrumcorrespond tobound statesin quantum mechanics, while acontinuous spectrumis associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

An important, notable development is theinverse scattering transform,central to the solution of manyexactly solvable models.

Theoretical physics

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Top: thereal partof aplane wavetravelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk ofindex of refractionhigher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact.

Inmathematical physics,scattering theoryis a framework for studying and understanding the interaction or scattering of solutions topartial differential equations.Inacoustics,the differential equation is thewave equation,and scattering studies how its solutions, thesound waves,scatter from solid objects or propagate through non-uniform media (such as sound waves, insea water,coming from asubmarine). In the case of classicalelectrodynamics,the differential equation is again the wave equation, and the scattering oflightorradio wavesis studied. Inparticle physics,the equations are those ofQuantum electrodynamics,Quantum chromodynamicsand theStandard Model,the solutions of which correspond tofundamental particles.

In regularquantum mechanics,which includesquantum chemistry,the relevant equation is theSchrödinger equation,although equivalent formulations, such as theLippmann-Schwinger equationand theFaddeev equations,are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and protons. The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and decays occurring. There are two predominant techniques of finding solutions to scattering problems:partial wave analysis,and theBorn approximation.

Electromagnetics

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AFeynman diagramof scattering between two electrons by emission of a virtualphoton

Electromagnetic wavesare one of the best known and most commonly encountered forms of radiation that undergo scattering.[16]Scattering of light and radio waves (especially inradar) is particularly important. Several different aspects of electromagnetic scattering are distinct enough to have conventional names. Major forms of elastic light scattering (involving negligible energy transfer) areRayleigh scatteringandMie scattering.Inelastic scattering includesBrillouin scattering,Raman scattering,inelasticX-rayscattering andCompton scattering.

Light scattering is one of the two major physical processes that contribute to the visible appearance of most objects, the other being absorption. Surfaces described aswhiteowe their appearance to multiple scattering of light by internal or surface inhomogeneities in the object, for example by the boundaries of transparent microscopic crystals that make up a stone or by the microscopic fibers in a sheet of paper. More generally, thegloss(orlustreorsheen) of the surface is determined by scattering. Highly scattering surfaces are described as being dull or having a matte finish, while the absence of surface scattering leads to a glossy appearance, as with polished metal or stone.

Spectral absorption, the selective absorption of certain colors, determines the color of most objects with some modification byelastic scattering.The apparent blue color ofveinsin skin is a common example where both spectral absorption and scattering play important and complex roles in the coloration. Light scattering can also create color without absorption, often shades of blue, as with the sky (Rayleigh scattering), the human blueiris,and the feathers of some birds (Prum et al. 1998). However, resonant light scattering innanoparticlescan produce many different highly saturated and vibrant hues, especially whensurface plasmon resonanceis involved (Roqué et al. 2006).[17][18]

Models of light scattering can be divided into three domains based on a dimensionless size parameter,αwhich is defined as: whereπDpis the circumference of a particle andλis the wavelength of incident radiation in the medium. Based on the value ofα,these domains are:

  • α≪ 1:Rayleigh scattering(small particle compared to wavelength of light);
  • α≈ 1:Mie scattering(particle about the same size as wavelength of light, valid only for spheres);
  • α≫ 1: geometric scattering (particle much larger than wavelength of light).

Rayleigh scatteringis a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive indexes, such as a particle, bubble, droplet, or even a density fluctuation. This effect was first modeled successfully byLord Rayleigh,from whom it gets its name. In order for Rayleigh's model to apply, the sphere must be much smaller in diameter than thewavelength(λ) of the scattered wave; typically the upper limit is taken to be about 1/10 the wavelength. In this size regime, the exact shape of the scattering center is usually not very significant and can often be treated as a sphere of equivalent volume. The inherent scattering that radiation undergoes passing through a pure gas is due to microscopic density fluctuations as the gas molecules move around, which are normally small enough in scale for Rayleigh's model to apply. This scattering mechanism is the primary cause of the blue color of the Earth's sky on a clear day, as the shorter blue wavelengths of sunlight passing overhead are more strongly scattered than the longer red wavelengths according to Rayleigh's famous 1/λ4relation. Along with absorption, such scattering is a major cause of the attenuation of radiation by theatmosphere.[19]The degree of scattering varies as a function of the ratio of the particle diameter to the wavelength of the radiation, along with many other factors includingpolarization,angle, andcoherence.[20]

For larger diameters, the problem of electromagnetic scattering by spheres was first solved byGustav Mie,and scattering by spheres larger than the Rayleigh range is therefore usually known asMie scattering.In the Mie regime, the shape of the scattering center becomes much more significant and the theory only applies well to spheres and, with some modification,spheroidsandellipsoids.Closed-form solutions for scattering by certain other simple shapes exist, but no general closed-form solution is known for arbitrary shapes.

Both Mie and Rayleigh scattering are considered elastic scattering processes, in which the energy (and thus wavelength and frequency) of the light is not substantially changed. However, electromagnetic radiation scattered by moving scattering centers does undergo aDoppler shift,which can be detected and used to measure the velocity of the scattering center/s in forms of techniques such aslidarandradar.This shift involves a slight change in energy.

At values of the ratio of particle diameter to wavelength more than about 10, the laws ofgeometric opticsare mostly sufficient to describe the interaction of light with the particle. Mie theory can still be used for these larger spheres, but the solution often becomes numerically unwieldy.

For modeling of scattering in cases where the Rayleigh and Mie models do not apply such as larger, irregularly shaped particles, there are many numerical methods that can be used. The most common arefinite-element methodswhich solveMaxwell's equationsto find the distribution of the scattered electromagnetic field. Sophisticated software packages exist which allow the user to specify the refractive index or indices of the scattering feature in space, creating a 2- or sometimes 3-dimensional model of the structure. For relatively large and complex structures, these models usually require substantial execution times on a computer.

Electrophoresisinvolves the migration ofmacromoleculesunder the influence of an electric field.[21]Electrophoretic light scatteringinvolves passing an electric field through a liquid which makes particles move. The bigger the charge is on the particles, the faster they are able to move.[22]

See also

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References

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  1. ^Newton, Isaac (1665). "A letter of Mr. Isaac Newton Containing his New Theory About Light and Colours".Philosophical Transactions.6.Royal Society of London: 3087.
  2. ^Herschel, William (1800). "Experiments on the Solar, and on the Terrestrial Rays that Occasion Heat".Philosophical Transactions.XC.Royal Society of London: 770.
  3. ^Tyndall, John (1874)."On the Atmosphere as a Vehicle of Sound".Philosophical Transactions of the Royal Society of London.164:221.Bibcode:1874RSPT..164..183T.JSTOR109101.
  4. ^Merritt, Ernest (5 Oct 1898)."The Magnetic Deflection of Diffusely Reflected Cathode Rays".Electrical Review.33(14): 217.
  5. ^"Recent Work with Röntgen Rays".Nature.53(1383): 613–616. 30 Apr 1896.Bibcode:1896Natur..53..613..doi:10.1038/053613a0.S2CID4023635.
  6. ^Rutherford, E.(1911). "The Scattering of α and β rays by Matter and the Structure of the Atom".Philosophical Magazine.6:21.
  7. ^Seinfeld,John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc.ISBN0-471-82857-2
  8. ^Colton, David; Rainer Kress (1998).Inverse Acoustic and Electromagnetic Scattering Theory.Springer.ISBN978-3-540-62838-5.
  9. ^Nachtmann, Otto (1990).Elementary Particle Physics: Concepts and Phenomena.Springer-Verlag. pp. 80–93.ISBN3-540-50496-6.
  10. ^"Zodiacal Glow Lightens Paranal Sky".ESO Picture of the Week.European Southern Observatory.Retrieved2 December2013.
  11. ^Gonis, Antonios; William H. Butler (1999).Multiple Scattering in Solids.Springer.ISBN978-0-387-98853-5.
  12. ^Gonis, Antonios; William H. Butler (1999).Multiple Scattering in Solids.Springer.ISBN978-0-387-98853-5.
  13. ^Stover, John C. (1995).Optical Scattering: Measurement and Analysis.SPIE Optical Engineering Press.ISBN978-0-8194-1934-7.
  14. ^R. F. Egerton (1996)Electron energy-loss spectroscopy in the electron microscope(Second Edition, Plenum Press, NY)ISBN0-306-45223-5
  15. ^Ludwig Reimer (1997)Transmission electron microscopy: Physics of image formation and microanalysis(Fourth Edition, Springer, Berlin)ISBN3-540-62568-2
  16. ^Colton, David; Rainer Kress (1998).Inverse Acoustic and Electromagnetic Scattering Theory.Springer.ISBN978-3-540-62838-5.
  17. ^Bohren, Craig F.; Donald R. Huffman (1983).Absorption and Scattering of Light by Small Particles.Wiley.ISBN978-0-471-29340-8.
  18. ^Roqué, Josep; J. Molera; P. Sciau; E. Pantos; M. Vendrell-Saz (2006). "Copper and silver nanocrystals in lustre lead glazes: development and optical properties".Journal of the European Ceramic Society.26(16): 3813–3824.doi:10.1016/j.jeurceramsoc.2005.12.024.
  19. ^Seinfeld,John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc.ISBN0-471-82857-2
  20. ^Prum, Richard O.;Rodolfo H. Torres;Scott Williamson; Jan Dyck (1998). "Coherent light scattering by blue feather barbs".Nature.396(6706): 28–29.Bibcode:1998Natur.396...28P.doi:10.1038/23838.S2CID4393904.
  21. ^"Understanding Electrophoretic Light Scattering".Wyatt Technology.
  22. ^"Light Scattering".Malvern Panalytical.
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