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{{Quantum book backlink|Quantum dynamics and evolution}}

In physics, '''scattering''' is a wide range of physical processes where moving particles or radiation of some form, such as [[light]] or [[sound]], are forced to deviate from a straight [[trajectory]] by 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 the [[law of reflection]]. Reflections of radiation that undergo scattering are often called ''[[diffuse reflection]]s'' and unscattered reflections are called ''[[specular]]'' (mirror-like) reflections. Originally, the term was confined to light scattering, going back at least as far as [[Isaac Newton]] in the 17th century.<ref>{{cite journal |last1=Newton |first1=Isaac |title=A letter of Mr. Isaac Newton Containing his New Theory About Light and Colours |journal=Philosophical Transactions |date=1665 |volume=6 |page=3087 |publisher=Royal Society of London}}</ref> As more "ray"-like phenomena were discovered, the concept of scattering was extended, so that [[William Herschel]] could refer to the scattering of heat rays in 1800.<ref>{{cite journal |last1=Herschel |first1=William |title=Experiments on the Solar, and on the Terrestrial Rays that Occasion Heat |journal=Philosophical Transactions |date=1800 |volume=XC |page=770 |publisher=Royal Society of London}}</ref> [[John Tyndall]] later noted the connection between light scattering and acoustic scattering in the 19th century.<ref>{{cite journal |last1=Tyndall |first1=John |title=On the Atmosphere as a Vehicle of Sound |journal=Philosophical Transactions of the Royal Society of London |date=1874 |volume=164 |page=221 |jstor=109101 |bibcode=1874RSPT..164..183T}}</ref> Near the end of the 19th century, the scattering of cathode rays and X-rays was observed and discussed, and with the discovery of subatomic particles and the development of quantum theory, the meaning 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.<ref>{{cite journal |last1=Merritt |first1=Ernest |title=The Magnetic Deflection of Diffusely Reflected Cathode Rays |journal=Electrical Review |date=5 Oct 1898 |volume=33 |issue=14 |page=217}}</ref><ref>{{cite journal |title=Recent Work with Röntgen Rays |journal=Nature |date=30 Apr 1896 |volume=53 |issue=1383 |pages=613–616 |doi=10.1038/053613a0 |bibcode=1896Natur..53..613. |s2cid=4023635}}</ref><ref>{{cite journal|first=E. |last=Rutherford |author-link=Ernest Rutherford |title=The Scattering of α and β rays by Matter and the Structure of the Atom |journal=Philosophical Magazine |volume=6 |page=21 |date=1911}}</ref>

Scattering can refer to the consequences of [[particle collision|particle-particle collisions]] between molecules, atoms, [[electron]]s, photons and other particles. Examples include [[cosmic ray]] scattering in the Earth's upper atmosphere, particle collisions inside [[particle accelerator]]s, electron scattering by gas atoms in fluorescent lamps, and [[neutron scattering]] inside [[nuclear reactor]]s.<ref>[[John H. Seinfeld|Seinfeld]], John H.; Pandis, Spyros N. (2006). ''Atmospheric Chemistry and Physics - From Air Pollution to Climate Change'' (2nd Ed.). John Wiley and Sons, Inc. {{ISBN|0-471-82857-2}}</ref>

The types of non-uniformities which can cause scattering, sometimes known as ''scatterers'' or ''scattering centers'', are too numerous to list, but a small sample includes [[particle]]s, [[Liquid bubble|bubble]]s, [[droplet]]s, [[density]] fluctuations in [[fluid]]s, [[crystallite]]s in [[polycrystal]]line solids, defects in [[monocrystal]]line solids, [[surface roughness]], [[cell (biology)|cell]]s in organisms, and textile [[fiber]]s in 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 of [[scattering theory]].

Scattering is quantified using many different concepts, including [[Cross section (physics)|scattering cross section]] (σ), [[attenuation coefficient]]s, the [[bidirectional scattering distribution function]] (BSDF), [[S-matrix|S-matrices]], and [[mean free path]].
[[File:Wine_glass_in_LCD_projectors_beam-y.jpg|thumb|400px|A wine glass in an LCD projector beam causes light to scatter, producing complex patterns due to refraction, diffraction, and interference.]]
== Classical and quantum scattering ==
In classical physics, scattering is often described in terms of trajectories and collisions. Light scattering by small particles leads to phenomena such as [[Rayleigh scattering]], which explains why the sky appears blue. In the classical picture, scattering may arise when waves or particles encounter irregularities in a material, such as density changes in a fluid, defects in a crystal, or roughness on a surface.

In quantum mechanics, scattering is described using [[wavefunction]]s and [[probability amplitude]]s. Instead of definite trajectories, particles are represented by waves that interact with a potential. The outcome of a scattering process is characterized by quantities such as the [[cross section]] and the [[scattering amplitude]]. In particle physics, the quantum interaction and scattering of fundamental particles is described by the [[S-Matrix]], introduced and developed by [[John Archibald Wheeler]] and [[Werner Heisenberg]].<ref>{{cite book |last1=Nachtmann |first1=Otto |title=Elementary Particle Physics: Concepts and Phenomena |publisher=Springer-Verlag |date=1990 |pages=80–93 |isbn=3-540-50496-6 }}</ref>

Scattering may also be classified as ''elastic'' or ''inelastic''. In elastic scattering, the internal states of the scattering particles do not change, and they emerge unchanged from the interaction. In inelastic scattering, by contrast, internal energy changes occur, which may excite atoms, ionize them, or even lead to the annihilation of some particles and the creation of entirely new ones. A well-known example in particle physics is [[deep inelastic scattering]], which has been crucial in probing the internal structure of hadrons.

== Single and multiple scattering ==
When radiation is only scattered by one localized scattering center, this is called ''single scattering''. It is more common that scattering centers are grouped together; in such cases, radiation may scatter many times, in what is known as ''multiple scattering''.<ref>{{cite book |last= Gonis |first= Antonios |author2=William H. Butler |title= Multiple Scattering in Solids |publisher= [[Springer Science+Business Media|Springer]] |year= 1999 |isbn= 978-0-387-98853-5 }}</ref>

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 often 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 with [[diffusion theory]].<ref>{{cite book |last= Gonis |first= Antonios |author2=William H. Butler |title= Multiple Scattering in Solids |publisher= [[Springer Science+Business Media|Springer]] |year= 1999 |isbn= 978-0-387-98853-5 }}</ref>

Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome often appears random to an observer. This type of scattering is exemplified by an electron being fired at an atomic nucleus: the exact position of the atom relative to the electron path is unknown, so the exact trajectory after the collision cannot be predicted. Single scattering is therefore often described by probability distributions.

With multiple scattering, the randomness of individual interactions tends to be averaged out by a large number of events, so that the final path of the radiation appears to be a more deterministic distribution of intensity. A familiar example is a [[light beam]] passing through thick [[fog]]. Multiple scattering is highly analogous to [[diffusion]], and the terms ''multiple scattering'' and ''diffusion'' are interchangeable in many contexts. Optical elements designed to produce multiple scattering are therefore known as ''diffusers''.<ref>{{cite book |last= Stover |first= John C. |title= Optical Scattering: Measurement and Analysis |publisher= SPIE Optical Engineering Press |year= 1995 |isbn= 978-0-8194-1934-7 }}</ref>

Not all single scattering is random, however. A well-controlled laser beam can be positioned to scatter from a microscopic particle with a nearly deterministic outcome. Similarly, multiple scattering can sometimes have random outcomes, especially for coherent radiation. The random fluctuations in the multiply scattered intensity of coherent radiation are called [[speckle pattern|speckle]]s. [[Coherent backscattering]], an enhancement of [[backscattering]] that occurs when coherent radiation is multiply scattered by a random medium, is usually attributed to [[weak localization]].

== Scattering theory ==
'''Scattering theory''' is a framework for studying and understanding the scattering of [[wave]]s and [[elementary particle]]s. More precisely, it concerns how solutions of [[partial differential equations]], propagating freely in the distant past, come together and interact with one another or with a boundary or potential, and then propagate away again into the distant future.

Wave scattering corresponds to the interaction of a wave with some material object, for instance sunlight scattered by raindrops to form a rainbow. Scattering also includes the interaction of billiard balls on a table, the [[Rutherford scattering]] of [[alpha particle]]s by atomic nuclei, the [[Bragg scattering]] of electrons and X-rays by a cluster of atoms, and the [[inelastic scattering]] of a fission fragment as it traverses a thin foil.

The '''direct scattering problem''' is the problem of determining the distribution of scattered radiation or particle flux from the characteristics of the scatterer. The [[inverse scattering problem]] is the problem of determining the characteristics of an object, such as its shape or internal constitution, from measurement data of radiation or particles scattered from it.

In regular [[quantum mechanics]], the relevant equation is the [[Schrödinger equation]], although equivalent formulations such as the [[Lippmann-Schwinger equation]] and the [[Faddeev equation]]s are also used. The solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and protons that come together from large distances, interact, and then move apart again. These solutions reveal the directions in which the products are most likely to travel, how quickly they move, and the probabilities of reactions, creations, and decays. Two predominant techniques for finding solutions to scattering problems are [[partial wave analysis]] and the [[Born approximation]].

Key concepts include:
* The [[scattering matrix]] (S-matrix), which relates incoming and outgoing states.
* The [[differential cross section]], describing the angular distribution of scattered particles.
* The [[phase shift]], which encodes how the wave is altered by the interaction.
* The [[inverse scattering problem]], in which properties of a system are inferred from scattering data.
* The [[mean free path]] and attenuation coefficients, which quantify how scattering reduces an unscattered beam.

=== Attenuation due to scattering ===
When the target is a set of many scattering centers whose relative positions vary unpredictably, it is useful to describe the decrease of an unscattered beam by an attenuation equation. In the simplest case, if particles are removed from the unscattered beam at a rate proportional to the incident intensity <math>I</math>, then

<math display="block">\frac{dI}{dx}=-QI</math>

where ''Q'' is an interaction coefficient and ''x'' is the distance traveled in the target.

This has solutions of the form

<math display="block">I = I_o e^{-Q \Delta x} = I_o e^{-\frac{\Delta x}{\lambda}} = I_o e^{-\sigma (\eta \Delta x)} = I_o e^{-\frac{\rho \Delta x}{\tau}}</math>

where ''I''<sub>o</sub> is the initial flux, λ is the interaction [[mean free path]], σ is the [[cross section (physics)|cross section]], η is the number of targets per unit volume, ρ is the target mass density, and τ is the density mean free path. These quantities provide different but related ways of measuring attenuation in scattering systems.

== Electromagnetic scattering ==
[[Electromagnetic radiation|Electromagnetic waves]] are among the most familiar forms of radiation that undergo scattering.<ref>{{cite book |last= Colton |first= David|author2=Rainer Kress | title= Inverse Acoustic and Electromagnetic Scattering Theory |publisher= [[Springer Science+Business Media|Springer]] |year= 1998 |isbn= 978-3-540-62838-5 }}</ref> Scattering of light and radio waves is particularly important in optics and radar. Major forms of elastic light scattering are [[Rayleigh scattering]] and [[Mie theory|Mie scattering]], while inelastic scattering includes [[Brillouin scattering]], [[Raman scattering]], inelastic X-ray scattering, and [[Compton 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 as white owe their appearance to multiple scattering of light by internal or surface inhomogeneities in the object, for example by microscopic crystals in stone or by the fibers in paper. More generally, the gloss or sheen of a surface is determined by scattering: highly scattering surfaces tend to appear dull or matte, while the absence of surface scattering leads to a glossy appearance.

Spectral absorption determines much of the color of objects, but scattering often modifies or even creates color. The blue color of the sky is a classic result of [[Rayleigh scattering]], in which shorter wavelengths are scattered more strongly than longer ones. Light scattering can also be responsible for the appearance of the human blue iris and the structural colors of certain bird feathers. For larger particles, the appropriate description shifts from Rayleigh scattering to [[Mie scattering]], and for particles much larger than the wavelength of light, the laws of geometric optics often become sufficient.

== Applications ==
Scattering and scattering theory are significant in many areas of physics and engineering. Important examples include radar sensing, [[medical ultrasound]], [[semiconductor wafer]] inspection, [[polymerization]] process monitoring, acoustic tiling, free-space communications, and [[computer-generated imagery]].<ref>{{cite book |last= Colton |first= David|author2=Rainer Kress |title= Inverse Acoustic and Electromagnetic Scattering Theory |publisher= [[Springer Science+Business Media|Springer]] |year= 1998 |isbn= 978-3-540-62838-5 }}</ref>

Particle-particle scattering theory is especially important in [[particle physics]], [[atomic, molecular, and optical physics]], [[nuclear physics]], and [[astrophysics]]. Experimental scattering techniques such as [[electron scattering]] and [[neutron scattering]] are essential tools for probing the microscopic structure of matter. In condensed matter and materials science, scattering methods are used to study defects, transport, and collective behavior. In atmospheric physics and optics, scattering determines visibility, color, radiative transfer, and the propagation of light through clouds, fog, and aerosols.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Scattering theory|1}}

Physics:Quantum Scattering theory - Revision history

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