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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 reflections 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.^[1] 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.^[2] John Tyndall later noted the connection between light scattering and acoustic scattering in the 19th century.^[3] 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.^[4]^[5]^[6]

Scattering can refer to the consequences of particle-particle collisions between molecules, atoms, electrons, photons and other particles. Examples include cosmic ray scattering in the Earth's upper atmosphere, particle collisions inside particle accelerators, electron scattering by gas atoms in fluorescent lamps, and neutron scattering inside nuclear reactors.^[7]

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 particles, bubbles, droplets, density fluctuations in fluids, crystallites in polycrystalline solids, defects in monocrystalline solids, surface roughness, cells in organisms, and textile fibers 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 scattering cross section (σ), attenuation coefficients, the bidirectional scattering distribution function (BSDF), S-matrices, and mean free path.

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 wavefunctions and probability amplitudes. 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.^[8]

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.^[9]

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.^[10]

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.^[11]

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 speckles. 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 waves and elementary particles. 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 particles 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 equations 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 $I$ , then

$\frac{d I}{d x} = - Q I$

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

This has solutions of the form

$I = I_{o} e^{- Q Δ x} = I_{o} e^{- \frac{Δ x}{λ}} = I_{o} e^{- σ (η Δ x)} = I_{o} e^{- \frac{ρ Δ x}{τ}}$

where I_o is the initial flux, λ is the interaction mean free path, σ is the 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 waves are among the most familiar forms of radiation that undergo scattering.^[12] Scattering of light and radio waves is particularly important in optics and radar. Major forms of elastic light scattering are Rayleigh scattering and 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.^[13]

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.

Table of contents (137 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Scattering theory

Classical and quantum scattering

Single and multiple scattering

Scattering theory

Attenuation due to scattering

Electromagnetic scattering

Applications

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