Editing Scattering (section)

== Theory ==
{{More citations needed|section|date=August 2023}}
Scattering theory is a framework for studying and understanding the scattering of [[wave]]s and [[Elementary particle|particles]]. 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 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.

===Attenuation due to scattering===
[[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 number of particles per unit area per unit time (<math>I</math>), 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&nbsp;≡&nbsp;''x''&nbsp;&minus;&nbsp;''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'' =&nbsp;1/''&lambda;'' =&nbsp;''&eta;&sigma;'' =&nbsp;''&rho;/&tau;'', as shown in the figure at left.

In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm<sup>&minus;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>&minus;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.

===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.

===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.  Solutions with a [[discrete spectrum (physics)|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.