In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
,
scattering theory is a framework for studying and understanding the
scatteringScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
of
wavesThe WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...
and
particlesIn 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 up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance
sunlightSunlight, in the broad sense, is the total spectrum of the electromagnetic radiation given off by the Sun. On Earth, sunlight is filtered through the atmosphere, and the solar radiation is obvious as daylight when the Sun is above the horizon. Near the poles in summer, the days are longer and the...
scattered by rain drops to form a
rainbowA rainbow is an optical and meteorological phenomenon that causes a spectrum of light to appear in the sky when the Sun shines onto droplets of moisture in the Earth's atmosphere...
. Scattering also includes the interaction of billiard balls on a table, the
Rutherford scatteringIn physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1909, and led to the development of the Rutherford model of the atom, and eventually to the Bohr model. It is now exploited by the materials analytical technique Rutherford backscattering...
(or angle change) of
alpha particleAlpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus; hence, it can be written as or . They have a net spin of zero, and normally a total energy of about 5 MeV...
s by
goldGold is a chemical element with the symbol Au and an atomic number of 79. It has been a highly sought-after precious metal for coinage, jewelry, and other arts since the beginning of recorded history. The metal occurs as nuggets or grains in rocks, in veins and in alluvial deposits. Gold is...
nucleiThe nucleus is the very dense region consisting of nucleons at the center of an atom. Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus, with a very small contribution from the orbiting electrons....
, 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 determining the distribution of scattered radiation/particle flux basing on the characteristics of the
scattererScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
.
The
inverse scattering problemIn physics, in the area of scattering theory, the inverse scattering problem is the problem of determining the characteristics of an object from measurement data of radiation or particles scattered from the object....
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
radiolocationRadar is an object detection system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR was coined in 1941 as an acronym for RAdio Detection And...
, the problem has found vast number of applications, such as
echolocationAcoustic location is the art and science of using sound to determine the distance and direction of something. Location can be done actively or passively, and can take place in gases , liquids , and in solids .* Active acoustic location involves the creation of sound in order to produce an echo,...
, geophysical survey,
nondestructive testingNondestructive testing is a wide group of analysis techniques used in science and industry to evaluate the properties of a material, component or system without causing damage. Because NDT does not permanently alter the article being inspected, it is a highly-valuable technique that can save both...
,
medical imagingMedical imaging refers to the techniques and processes used to create images of the human body for clinical purposes or medical science .As a discipline and in its widest sense, it is part of biological imaging and incorporates...
and
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, 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
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 equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
has solutions of the form:
where I
o is the initial flux, path length Δx≡x-x
o, the second equality defines an interaction
mean free pathIn physics the mean free path of a particle is the average distance covered by a particle between successive impacts.. Alternatively, it is the distance at which the intensity of particles drops by 1/e.-Derivation:...
λ, the third uses the number of targets per unit volume η to define an area
cross-sectionIn nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles.When particles are thrown against a foil made of a certain substance, the cross section is a hypothetical area measure around the target particles that represents a...
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
-1) is variously called
opacityOpacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc...
, absorption coefficient, and
attenuation coefficientThe attenuation coefficient is a basic quantity used in calculations of the penetration of materials by quantum particles or other energy beams. It is a measure of attenuation....
. In nuclear physics, area cross-sections (e.g. σ in
barnA barn is a unit of area. Originally used in nuclear physics for expressing the cross sectional area of nuclei and nuclear reactions, today it is used in all fields of high energy physics to express the cross sections of any scattering process...
s or units of 10
-24 cm
2), density mean free path (e.g. τ in grams/cm
2), and its reciprocal the
mass attenuation coefficientThe mass attenuation coefficient is a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per unit mass...
(e.g. in cm
2/gram) or
area per nucleon are all popular, while in electron microscopy the
inelastic mean free pathThe inelastic mean free path is an index of how far an electron can travel through a solid before losing energy.If a monochromatic primary beam of electrons is incident on a solid surface, the majority of incident electrons lose their energy because they interact strongly with matter, leading to...
(e.g. λ in nanometers) is often discussed instead.
In theoretical physics
In
mathematical physicsMathematical physics is the scientific discipline concerned with the interface of mathematics and physics. The Journal of Mathematical Physics defines it as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the...
,
scattering theory is a framework for studying and understanding the interaction or
scatteringScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
of solutions to
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s. In
acousticsAcoustics is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound . A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering...
, the differential equation is the
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
, 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
submarineA submarine is a watercraft capable of independent operation below the surface of the water. It differs from a submersible, which has only limited underwater capability...
). In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of
lightLight is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye ....
or
radio waveRadio Wave may refer to:*Radio frequency*Radio Wave 96.5, a radio station in Blackpool, UK...
s is studied. In
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
and
particle physicsParticle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called high energy physics, because many elementary particles do not occur under normal circumstances in nature, but can be created and detected...
, the equations are those of quantum electrodynamics
QEDQuantum electrodynamics is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons...
, quantum chromodynamics
QCDIn theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
and the
Standard ModelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
, the solutions of which correspond to fundamental particles. In
quantum chemistryQuantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. The description of the electronic behavior of atoms and molecules as pertaining to their reactivity is one of the applications of quantum...
, the solutions correspond to
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s and
moleculeA molecule is defined as an electrically neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds. Molecules are distinguished from polyatomic ions in this strict sense...
s, governed by the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
.
Elastic and inelastic scattering
The example of scattering in
quantum chemistryQuantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. The description of the electronic behavior of atoms and molecules as pertaining to their reactivity is one of the applications of quantum...
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 stateIn physics, a bound state is a composite that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is...
solutions of some differential equation. Thus, for example, the
hydrogen atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force. The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no...
corresponds to a solution to the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
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
ionizedIonization is the physical process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation ....
. Thus,
collisionA collision is an isolated event in which two or more moving bodies exert relatively strong forces on each other for a relatively short time.-Dynamics:Collisions involve forces...
s can be either
elasticAn 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. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms...
(the internal quantum states of the particles are not changed) or
inelasticAn inelastic collision is a collision in which kinetic energy is not conserved .In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas or liquid rarely experience perfectly...
(the internal quantum states of the particles are changed). From the experimental viewpoint the observable quantity is the
cross sectionIn nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles.When particles are thrown against a foil made of a certain substance, the cross section is a hypothetical area measure around the target particles that represents a...
. From the theoretical viewpoint the key quantity is the
S matrixIn physics, the scattering matrix relates the initial state and the final state for an interaction of particles. It is used in quantum mechanics, scattering theory and quantum field theory....
.
Topics in physics
According to the optics classification of the
Optical Society of AmericaThe Optical Society of America is a scientific society dedicated to advancing the study of light—optics and photonics—in theory and application, by means of worldwide research, scientific publishing, conferences and exhibitions, partnership with industry, and the education of new generations of...
this field consists of the following topics:
The mathematical framework
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
. 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
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s. Frequently, the means to the solution requires the study of the
spectrumA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
of an
operatorIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
on the manifold. As a result, the solutions often have a spectrum that can be identified with a
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, and scattering is described by a certain map, the
S matrixIn physics, the scattering matrix relates the initial state and the final state for an interaction of particles. It is used in quantum mechanics, scattering theory and quantum field theory....
, on Hilbert spaces. Spaces with a
discrete spectrumIn physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a discrete spectrum if its spectrum consists of isolated points. If the spectrum of an operator is not discrete, we say that it is a continuous...
correspond to
bound stateIn physics, a bound state is a composite that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is...
s in quantum mechanics, while a
continuous spectrumIn physics, continuous spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals. The term is derived from the use of the word spectrum to describe the 'ghost-like' rainbow which appears when white light is shone through a clear...
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 transformIn mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...
, central to the solution of many exactly solvable models.