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	~~46 year old Dental Hygienist Cadotte~~ from ~~York~~, ~~loves learn~~, ~~new launch property singapore~~ and ~~fossils~~. ~~Discovers~~ the ~~charm~~ in ~~touring destinations around~~ the ~~entire world~~, ~~recently only returning~~ from ~~Palau de la Música Catalana~~ and ~~Hospital de Sant Pau~~.<br><br>~~my page~~; [http://www.~~mydadhasrights~~.~~com~~/~~member~~.~~php?u~~=~~821113~~-~~M7498 The Skywoods condo price~~]		[[Image:Scattering theory illust.png\|right\|thumb\|Top: the [[real part]] of a [[plane wave]] travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of [[index of refraction]] higher 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.]]
			In [[mathematics]] and [[physics]], '''scattering theory''' is a framework for studying and understanding the [[scattering]] of [[waves]] and [[Elementary particle\|particles]]. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance [[sunlight]] scattered by [[rain drop]]s to form a [[rainbow]]. Scattering also includes the interaction of [[billiard balls]] on a table, the [[Rutherford scattering]] (or angle change) of [[alpha particle]]s by [[gold]] [[atomic nucleus\|nuclei]], the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the inelastic scattering of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of [[partial differential equations]], propagating freely "in the distant past", come together and interact with one another or with a [[boundary condition]], and then propagate away "to the distant future". The '''direct scattering problem''' is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the [[scattering\|scatterer]]. The [[inverse scattering problem]] is 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.

			Since its early statement for [[radar\|radiolocation]], the problem has found vast number of applications, such as [[acoustic location\|echolocation]], [[geophysical]] survey, [[nondestructive testing]], [[medical imaging]] and [[quantum field theory]], to name just a few.

			==Conceptual underpinnings==

			The concepts used in scattering theory go by different names in
			different fields. The object of this section is to point the reader
			to common threads.

			===Composite targets and range equations===

			[[Image:Xsection2.png\|288px\|thumb\|left\|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 flux <math>I</math> of particles per unit area per unit time, i.e. that

			:<math> \frac{dI}{dx}=-QI \,\!</math>

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

			The above ordinary first-order [[differential equation]] has solutions of the form:

			: <math>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, path length Δx ≡ ''x'' − ''x''<sub>o</sub>, the second equality defines an interaction [[mean free path]] λ, the third uses the number of targets per unit volume η to define an area [[cross section (physics)\|cross-section]] σ, and the last uses the target mass density ρ to define a density mean free path τ. Hence one converts between these quantities via ''Q'' = 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<sup>−1</sup>) is variously called [[opacity (optics)\|opacity]], [[absorption coefficient]], and [[attenuation coefficient]]. In nuclear physics, area cross-sections (e.g. σ in [[barn (unit)\|barn]]s or units of 10<sup>−24</sup> cm<sup>2</sup>), density mean free path (e.g. τ in grams/cm<sup>2</sup>), and its reciprocal the [[mass attenuation coefficient]] (e.g. in cm<sup>2</sup>/gram) or ''area per nucleon'' are all popular, while in electron microscopy the [[inelastic mean free path]]<ref>R. F. Egerton (1996) ''Electron energy-loss spectroscopy in the electron microscope'' (Second Edition, Plenum Press, NY) ISBN 0-306-45223-5</ref> (e.g. λ in nanometers) is often discussed<ref>Ludwig Reimer (1997) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Fourth Edition, Springer, Berlin) ISBN 3-540-62568-2</ref> instead.

			==In theoretical physics==
			In [[mathematical physics]], '''scattering theory''' is a framework for studying and understanding the interaction or [[scattering]] of solutions to [[partial differential equation]]s. In [[acoustics]], the differential equation is the [[wave equation]], and scattering studies how its solutions, the [[sound wave]]s, scatter from solid objects or propagate through non-uniform media (such as sound waves, in [[sea water]], coming from a [[submarine]]). In the case of classical [[electrodynamics]], the differential equation is again the wave equation, and the scattering of [[light]] or [[radio wave]]s is studied. In [[particle physics]], the equations are those of [[Quantum electrodynamics]], [[Quantum chromodynamics]] and the [[Standard Model]], the solutions of which correspond to [[fundamental particle]]s.

			In regular [[quantum mechanics]], which includes [[quantum chemistry]], the relevant equation is the [[Schrödinger equation]], although equivalent formulations, such as the [[Lippmann-Schwinger equation]] and the Faddeev equations, are also largely used. The solutions 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 the [[Born approximation]].

			==Elastic and inelastic scattering==

			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 in [[quantum chemistry]] is 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 the [[bound state]] solutions of some differential equation. Thus, for example, the [[hydrogen atom]] corresponds to a solution to the [[Schrödinger equation]] with 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 even [[ionization\|ionized]], representing an inelastic scattering process.

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

			==The mathematical framework==
			In [[mathematics]], scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a [[differential equation]] is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of [[time]]. 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 on [[manifold]]s. Frequently, the means to the solution requires the study of the [[ Spectrum (functional analysis) \| spectrum]] of an [[operator theory\|operator]] on the manifold. As a result, the solutions often have a spectrum that can be identified with a [[Hilbert space]], and scattering is described by a certain map, the [[S matrix]], on Hilbert spaces. Spaces with a [[discrete spectrum]] correspond to [[bound state]]s in quantum mechanics, while a [[continuous spectrum]] is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

			An important, notable development is the [[inverse scattering transform]], central to the solution of many [[exactly solvable model]]s.
			==See also==
			{{div col}}
			*[[Backscattering]]
			*[[Brillouin scattering]]
			*[[Diffuse sky radiation]]
			*[[Extinction (astronomy)\|Extinction]]
			*[[Linewidth]]
			*[[Mie theory]]
			*[[Molecular scattering]]
			*[[Raman scattering]]
			*[[Rayleigh scattering]]
			*[[Scattering from rough surfaces]]
			*[[Scintillation (physics)]]
			{{div col end}}

			==Footnotes==
			{{reflist}}

			==References==
			* [http://www.mif.pg.gda.pl/homepages/slawek/epic/sem.htm Lectures of the European school on theoretical methods for electron and positron induced chemistry, Prague, Feb. 2005]
			* [http://fa.its.tudelft.nl/~koelink/dictaat-scattering.pdf E. Koelink, Lectures on scattering theory, Delft the Netherlands 2006]
			<!--
			{{Author = Koelink, E. \| Title = Lectures on scattering theory \| Url = http://fa.its.tudelft.nl/~koelink/dictaat-scattering.pdf\| Year = 2006 \|Place = Delft}}
			-->

			==External links==
			*[http://www.opticsinfobase.org/submit/ocis/OCIS_2007.pdf Optics Classification and Indexing Scheme (OCIS)], [[Optical Society of America]], 1997

			[[Category:Scattering theory\| ]]
			[[Category:Scattering, absorption and radiative transfer (optics)\| ]]
			[[Category:Quantum mechanics]]
			[[Category:Hilbert space]]

en>Lamro: link

2012-08-30T14:33:26Z

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46 year old Dental Hygienist Cadotte from York, loves learn, new launch property singapore and fossils. Discovers the charm in touring destinations around the entire world, recently only returning from Palau de la Música Catalana and Hospital de Sant Pau.<br><br>my page; [http://www.mydadhasrights.com/member.php?u=821113-M7498 The Skywoods condo price]

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en>Trappist the monk: /* History */Remove Lang-xx templates from CS1 citations; not COinS safe; using AWB

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en>Beagel: Fixing style/layout errors

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