In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Sunlight, in the broad sense is the total spectrum of the Electromagnetic radiation given off by the Sun. Rain is Liquid precipitation. On Earth it is the condensation of atmospheric Water vapor into drops heavy enough to fall often making it to A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun Scattering also includes the interaction of billiard balls on a table, the Rutherford scattering (or angle change) of alpha particles by gold 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. Billiard balls are used in Cue sports, such as Carom billiards, pool, and Snooker. In Physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1909 and led to the development of the orbital theory of the Alpha particles (named after and denoted by the first letter in the Greek alphabet, α consist of two Protons and two Neutrons bound together into a Gold (ˈɡoʊld is a Chemical element with the symbol Au (from its Latin name aurum) and Atomic number 79 The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom 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". In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints
The direct scattering problem is the problem determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from
The inverse scattering problem is the problem of determining the characteristics of an object (e. In Physics, in the area of Scattering theory, the inverse scattering problem is the problem of determining the characteristics of an object (its shape internal constitution g. , its shape, internal constitution) from measurement data of radiation or particles scattered from the object.
Since its early statement for radiolocation, the problem has found vast number of applications, such as echolocation, geophysical survey, nondestructive testing, medical imaging and quantum field theory, to name just a few. Radar is a system that uses electromagnetic waves to identify the range altitude direction or speed of both moving and fixed objects such as Aircraft, ships Geophysics, a major discipline of Earth sciences, is the study of the Earth by quantitative physical methods especially by seismic, electromagnetic Nondestructive testing ( NDT) also called nondestructive examination ( NDE) and nondestructive inspection ( NDI) is testing that does Medical imaging refers to the techniques and processes used to create Images of the human body (or parts thereof for clinical purposes ( Medical procedures seeking to In quantum field theory (QFT the forces between particles are mediated by other particles
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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
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 I of particles per unit area per unit time, i. e. that dI/dx = -QI 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:
where Io is the initial flux, path length Δx≡x-xo, 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 σ, and the last uses the target mass density ρ to define a density mean free path τ. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Physics the mean free path of a particle is the average distance covered by a particle ( Photon, Atom or Molecule) between subsequent impacts In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles 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-1) is variously called opacity, absorption coefficient, and attenuation coefficient. Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation especially visible Light. The absorption coefficient α is a property of a material It defines the extent to which a material absorbs energy for example that of Sound waves or Electromagnetic For attenuation coefficient as it applies to electromagnetic theory and telecommunications see propagation constant. In nuclear physics, area cross-sections (e. g. σ in barns or units of 10-24 cm2), density mean free path (e. A barn (symbol b) is a unit of Area. While the barn is not an SI unit it is accepted (although discouraged for use with the SI g. τ in grams/cm2), and its reciprocal the mass attenuation coefficient (e. For attenuation coefficient as it applies to electromagnetic theory and telecommunications see propagation constant. g. in cm2/gram) or area per nucleon are all popular, while in electron microscopy the inelastic mean free path[1] (e. The inelastic mean free path ( IMFP) is an index of how far an Electron can travel through a solid before losing energy g. λ in nanometers) is often discussed[2] 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 equations. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or propagate through non-uniform media (such as sound waves, in sea water, coming from a submarine). Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves An acoustic wave is a weak compression wave (meaning a small pressure change that moves at the Speed of sound. Seawater is Water from a Sea or Ocean. On average seawater in the world's oceans has a Salinity of about 3 A submarine is a Watercraft that can operate independently below water as distinct from a Submersible that has only limited underwater capability In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of light or radio waves is studied. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Radio waves are electromagnetic waves occurring on the Radio frequency portion of the Electromagnetic spectrum. In quantum mechanics and particle physics, the equations are those of quantum electrodynamics QED, quantum chromodynamics QCD and the Standard Model, the solutions of which correspond to fundamental particles. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made In quantum chemistry, the solutions correspond to atoms and molecules, governed by the Schrödinger equation. Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system
Elastic and inelastic scattering
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. Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in When two atoms are scattered off one another, one can understand them as being the bound state solutions of some differential equation. In Physics, a bound state is a composite of two or more building blocks ( particles or bodies) that behaves as a single object Thus, for example, the hydrogen atom corresponds to a solution to the Schrödinger equation with a negative inverse-power (i. A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system 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 ionized. Ionization is the physical process of converting an Atom or Molecule into an Ion by adding or removing charged particles such as Electrons Thus, collisions can be either elastic (the internal quantum states of the particles are not changed) or inelastic (the internal quantum states of the particles are changed). A collision is an isolated event in which two or more bodies (colliding bodies exert relatively strong forces on each other for a relatively short time elastic collision is a collision in which the total Kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision An inelastic collision is a Collision in which kinetic energy is not conserved (see elastic collision) From the experimental viewpoint the observable quantity is the cross section. In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles From the theoretical viewpoint the key quantity is the S matrix. Scattering matrix redirects here For the meaning in linear electrical networks see Scattering parameters.
Topics in physics
According to the optics classification of the Optical Society of America this field consists of the following topics:
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The mathematical framework
In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts. The Optical Society of America (OSA is a scientific society dedicated to advancing the study of Light — Optics and Photonics —in theory and application Diffuse sky radiation is Solar radiation reaching the Earth 's surface after having been scattered from the direct solar beam by Molecules Backscatter is the reflection of Waves particles or signals back to the direction they came from Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement Extinction is a term used in Astronomy to describe the absorption and Scattering of Electromagnetic radiation emitted by Astronomical objects In Physics, in the area of Scattering theory, the inverse scattering problem is the problem of determining the characteristics of an object (its shape internal constitution The spectral linewidth characterizes the width of a Spectral line, such as in the electromagnetic emission spectrum of an atom or the Frequency spectrum Mie theory, also called Lorenz-Mie theory or Lorenz-Mie-Debye theory, is a complete analytical solution of Maxwell's equations for the Scattering Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from Brillouin scattering, named for Léon Brillouin, occurs when Light in a medium (such as Water or a Crystal) interacts with time dependent Raman scattering or the Raman effect (pronounced — is the inelastic scattering of a Photon. Rayleigh scattering (named after Lord Rayleigh) is the elastic Scattering of Light or other electromagnetic radiation by particles much smaller Surface roughness scattering or interface roughness scattering is the elastic Scattering of a charged particle by an imperfect interface between two different Brillouin scattering, named for Léon Brillouin, occurs when Light in a medium (such as Water or a Crystal) interacts with time dependent Raman scattering or the Raman effect (pronounced — is the inelastic scattering of a Photon. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of 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 manifolds. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold. A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. In Mathematics, operator theory is the branch of Functional analysis which deals with Bounded linear operators and their properties 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. This article assumes some familiarity with Analytic geometry and the concept of a limit. Scattering matrix redirects here For the meaning in linear electrical networks see Scattering parameters. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. In Physics, discrete spectrum is a Finite set or a Countable set of Eigenvalues of an Operator. In Physics, a bound state is a composite of two or more building blocks ( particles or bodies) that behaves as a single object In Physics, continuous spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals 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 models. In Mathematics, the inverse scattering transform is a method for solving some non-linear Partial differential equations. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems.
References
Footnotes
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