1361:, are also largely used. The solutions of interest 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:
1645:) of the scattered wave; typically the upper limit is taken to be about 1/10 the wavelength. In this size regime, the exact shape of the scattering center is usually not very significant and can often be treated as a sphere of equivalent volume. The inherent scattering that radiation undergoes passing through a pure gas is due to microscopic density fluctuations as the gas molecules move around, which are normally small enough in scale for Rayleigh's model to apply. This scattering mechanism is the primary cause of the blue color of the Earth's sky on a clear day, as the shorter blue wavelengths of sunlight passing overhead are more strongly scattered than the longer red wavelengths according to Rayleigh's famous 1/
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534:. 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 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
1200:. 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".
598:. Speckle also occurs if multiple parts of a coherent wave scatter from different centers. In certain rare circumstances, multiple scattering may only involve a small number of interactions such that the randomness is not completely averaged out. These systems are considered to be some of the most difficult to model accurately.
1436:
1144:
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
1714:
to find the distribution of the scattered electromagnetic field. Sophisticated software packages exist which allow the user to specify the refractive index or indices of the scattering feature in space, creating a 2- or sometimes 3-dimensional model of the structure. For relatively large and complex
541:
Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome, which tends to depend strongly on the exact incoming trajectory, appears random to an observer. This type of scattering would be exemplified by an electron being fired at
1058:
542:
an atomic nucleus. In this case, the atom's exact position relative to the path of the electron is unknown and would be unmeasurable, so the exact trajectory of the electron after the collision cannot be predicted. Single scattering is therefore often described by probability distributions.
789:
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
782:
1683:
Both Mie and
Rayleigh scattering are considered elastic scattering processes, in which the energy (and thus wavelength and frequency) of the light is not substantially changed. However, electromagnetic radiation scattered by moving scattering centers does undergo a
1488:
owe their appearance to multiple scattering of light by internal or surface inhomogeneities in the object, for example by the boundaries of transparent microscopic crystals that make up a stone or by the microscopic fibers in a sheet of paper. More generally, the
1632:
is a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive indexes, such as a particle, bubble, droplet, or even a density fluctuation. This effect was first modeled successfully by
545:
With multiple scattering, the randomness of the interaction tends to be averaged out by a large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity. This is exemplified by a
1152:
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
256:
in 1911) and the development of quantum theory in the 20th century, the sense 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.
1501:) of the surface is determined by scattering. Highly scattering surfaces are described as being dull or having a matte finish, while the absence of surface scattering leads to a glossy appearance, as with polished metal or stone.
1512:
in skin is a common example where both spectral absorption and scattering play important and complex roles in the coloration. Light scattering can also create color without absorption, often shades of blue, as with the sky
888:
585:
Not all single scattering is random, however. A well-controlled laser beam can be exactly positioned to scatter off a microscopic particle with a deterministic outcome, for instance. Such situations are encountered in
1457:) is particularly important. Several different aspects of electromagnetic scattering are distinct enough to have conventional names. Major forms of elastic light scattering (involving negligible energy transfer) are
219:
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
593:
Similarly, multiple scattering can sometimes have somewhat random outcomes, particularly with coherent radiation. The random fluctuations in the multiply scattered intensity of coherent radiation are called
1706:
For modeling of scattering in cases where the
Rayleigh and Mie models do not apply such as larger, irregularly shaped particles, there are many numerical methods that can be used. The most common are
1581:
1703:
are mostly sufficient to describe the interaction of light with the particle. Mie theory can still be used for these larger spheres, but the solution often becomes numerically unwieldy.
862:
773:
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.
1270:
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.
248:, a pioneer in light scattering research, noted the connection between light scattering and acoustic scattering in the 1870s. Near the end of the 19th century, the scattering of
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1484:
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
2398:
Roqué, Josep; J. Molera; P. Sciau; E. Pantos; M. Vendrell-Saz (2006). "Copper and silver nanocrystals in lustre lead glazes: development and optical properties".
1653:. The degree of scattering varies as a function of the ratio of the particle diameter to the wavelength of the radiation, along with many other factors including
808:
1672:. In the Mie regime, the shape of the scattering center becomes much more significant and the theory only applies well to spheres and, with some modification,
1729:
involves passing an electric field through a liquid which makes particles move. The bigger the charge is on the particles, the faster they are able to move.
1196:
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
2606:
1453:
are one of the best known and most commonly encountered forms of radiation that undergo scattering. Scattering of light and radio waves (especially in
1053:{\displaystyle 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 }}},}
723:. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance (sunlight) scattered by
339:
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
2429:, John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc.
2117:, John H.; Pandis, Spyros N. (2006). Atmospheric Chemistry and Physics - From Air Pollution to Climate Change (2nd Ed.). John Wiley and Sons, Inc.
3159:
1680:. Closed-form solutions for scattering by certain other simple shapes exist, but no general closed-form solution is known for arbitrary shapes.
406:
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is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.
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is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. The
1089:Ï, and the last uses the target mass density Ï to define a density mean free path Ï. Hence one converts between these quantities via
530:. It is more common that scattering centers are grouped together; in such cases, radiation may scatter many times, in what is known as
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travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of
1504:
Spectral absorption, the selective absorption of certain colors, determines the color of most objects with some modification by
1169:. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even
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1637:, from whom it gets its name. In order for Rayleigh's model to apply, the sphere must be much smaller in diameter than the
790:"unscattered beam" at a uniform rate that is proportional to the incident number of particles per unit area per unit time (
175:
683:
240:
in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that
754:
of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of
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89:
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Lectures of the
European school on theoretical methods for electron and positron induced chemistry, Prague, Feb. 2005
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a scattering atom, or the complete annihilation of a scattering particle and the creation of entirely new particles.
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In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm) is variously called
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2739:
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2222:
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1726:
1688:, which can be detected and used to measure the velocity of the scattering center/s in forms of techniques such as
662:
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3006:
3001:
2724:
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are interchangeable in many contexts. Optical elements designed to produce multiple scattering are thus known as
601:
The description of scattering and the distinction between single and multiple scattering are tightly related to
236:(mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as
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Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units
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1192:, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a
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relation. Along with absorption, such scattering is a major cause of the attenuation of radiation by the
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487:
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the quantum interaction and scattering of fundamental particles is described by the
Scattering Matrix or
244:
could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800.
31:
1945:
Newton, Isaac (1665). "A letter of Mr. Isaac Newton
Containing his New Theory About Light and Colours".
252:(electron beams) and X-rays was observed and discussed. With the discovery of subatomic particles (e.g.
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that occurs when coherent radiation is multiply scattered by a random medium, is usually attributed to
359:
1964:
Herschel, William (1800). "Experiments on the Solar, and on the
Terrestrial Rays that Occasion Heat".
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Models of light scattering can be divided into three domains based on a dimensionless size parameter,
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scattering as well, where the targets tend to be macroscopic objects such as people or aircraft.
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on the manifold. As a result, the solutions often have a spectrum that can be identified with a
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is a wide range of physical processes where moving particles or radiation of some form, such as
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For larger diameters, the problem of electromagnetic scattering by spheres was first solved by
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750:, the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the
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is a framework for studying and understanding the interaction or scattering of solutions to
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1996:
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At values of the ratio of particle diameter to wavelength more than about 10, the laws of
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Some areas where scattering and scattering theory are significant include radar sensing,
161:
119:
2465:
2450:; Scott Williamson; Jan Dyck (1998). "Coherent light scattering by blue feather barbs".
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2000:
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or units of 10 cm), density mean free path (e.g. Ï in grams/cm), and its reciprocal the
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structures, these models usually require substantial execution times on a computer.
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When radiation is only scattered by one localized scattering center, this is called
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2096:(1911). "The Scattering of α and ÎČ rays by Matter and the Structure of the Atom".
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Scattering theory is a framework for studying and understanding the scattering of
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Transmission electron microscopy: Physics of image formation and microanalysis
2188:
1619:(particle about the same size as wavelength of light, valid only for spheres);
1525:
can produce many different highly saturated and vibrant hues, especially when
1310:, the differential equation is again the wave equation, and the scattering of
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3201:
3054:
2945:
2779:
2749:
2702:
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is the wavelength of incident radiation in the medium. Based on the value of
1522:
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1216:
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300:
287:
The types of non-uniformities which can cause scattering, sometimes known as
196:
1085:λ, the third uses the number of targets per unit volume η to define an area
3084:
2697:
2692:
1254:
453:
375:
245:
237:
1625:â« 1: geometric scattering (particle much larger than wavelength of light).
3122:
1919:
1228:
1189:
1180:" refers to a special kind of scattering experiment in particle physics.
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1121:
324:
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316:
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144:
1829:
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1669:
1665:
1638:
1462:
1407: in this section. Unsourced material may be challenged and removed.
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1295:
1263:
1170:
643: in this section. Unsourced material may be challenged and removed.
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272:
scattering in the Earth's upper atmosphere; particle collisions inside
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2008:
1984:
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E. Koelink, Lectures on scattering theory, Delft the
Netherlands 2006
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555:
433:
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362:. Particle-particle scattering theory is important in areas such as
224:. Reflections of radiation that undergo scattering are often called
2535:
Research group on light scattering and diffusion in complex systems
2534:
1904:
1220:
1204:
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410:
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358:
process monitoring, acoustic tiling, free-space communications and
296:
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1207:. Frequently, the means to the solution requires the study of the
397:
Scattering is quantified using many different concepts, including
1157:
solutions of some differential equation. Thus, for example, the
728:
308:
304:
2540:
Multiple light scattering from a photonic science point of view
2397:
1444:
58:
2024:"The Magnetic Deflection of Diffusely Reflected Cathode Rays"
1693:
1689:
1509:
1474:
1454:
1443:
of scattering between two electrons by emission of a virtual
1311:
587:
336:
312:
276:; electron scattering by gas atoms in fluorescent lamps; and
212:
208:
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of scattering between two electrons by emission of a virtual
2301:
Electron energy-loss spectroscopy in the electron microscope
781:
1197:
1165:
with a negative inverse-power (i.e., attractive
Coulombic)
744:
716:
1989:
551:
295:, are too numerous to list, but a small sample includes
2368:
1203:
Solutions to differential equations are often posed on
3417:
Scattering, absorption and radiative transfer (optics)
2506:
2342:
Inverse
Acoustic and Electromagnetic Scattering Theory
2245:
2216:
2136:
Inverse Acoustic and Electromagnetic Scattering Theory
1120:. In nuclear physics, area cross-sections (e.g. Ï in
2371:
Absorption and Scattering of Light by Small Particles
1542:
891:
819:
796:
2488:
1219:, and scattering is described by a certain map, the
2165:
Elementary Particle Physics: Concepts and Phenomena
1139:
1136:(e.g. λ in nanometers) is often discussed instead.
762:, and then propagate away "to the distant future".
2339:
2133:
1576:{\displaystyle \alpha =\pi D_{\text{p}}/\lambda ,}
1575:
1132:are all popular, while in electron microscopy the
1052:
856:
802:
2555:World directory of neutron scattering instruments
1696:. This shift involves a slight change in energy.
1609:(small particle compared to wavelength of light);
853:
420:
268:, photons and other particles. Examples include:
3378:
2565:Optics Classification and Indexing Scheme (OCIS)
2496:"Understanding Electrophoretic Light Scattering"
1353:, although equivalent formulations, such as the
1294:, and scattering studies how its solutions, the
1173:, representing an inelastic scattering process.
1070:is the initial flux, path length Îx âĄ
776:
731:. Scattering also includes the interaction of
407:bidirectional scattering distribution function
2600:
1850:Phase space measurement with forward modeling
1081:, the second equality defines an interaction
554:. Multiple scattering is highly analogous to
169:
2369:Bohren, Craig F.; Donald R. Huffman (1983).
2277:Optical Scattering: Measurement and Analysis
260:Scattering can refer to the consequences of
2246:Gonis, Antonios; William H. Butler (1999).
2217:Gonis, Antonios; William H. Butler (1999).
494:. Unsourced material may be challenged and
2607:
2593:
2092:
1725:under the influence of an electric field.
176:
162:
2162:
2069:
1985:"On the Atmosphere as a Vehicle of Sound"
1423:Learn how and when to remove this message
1238:An important, notable development is the
1183:
852:
703:Learn how and when to remove this message
514:Learn how and when to remove this message
1963:
1875:Resonances in scattering from potentials
1434:
1334:, the solutions of which correspond to
1253:
875:is the distance traveled in the target.
857:{\displaystyle {\frac {dI}{dx}}=-QI\,\!}
780:
432:is a faint, diffuse glow visible in the
424:
215:, are forced to deviate from a straight
187:
2401:Journal of the European Ceramic Society
2021:
1982:
1590:is the circumference of a particle and
1223:, on Hilbert spaces. Solutions with a
230:and unscattered reflections are called
14:
3379:
2274:
1944:
1249:
368:atomic, molecular, and optical physics
2614:
2588:
2340:Colton, David; Rainer Kress (1998).
2189:"Zodiacal Glow Lightens Paranal Sky"
2134:Colton, David; Rainer Kress (1998).
1405:adding citations to reliable sources
1376:
641:adding citations to reliable sources
612:
492:adding citations to reliable sources
459:
2323:(Fourth Edition, Springer, Berlin)
2303:(Second Edition, Plenum Press, NY)
2167:. Springer-Verlag. pp. 80â93.
1372:
1290:, the differential equation is the
24:
2414:10.1016/j.jeurceramsoc.2005.12.024
2279:. SPIE Optical Engineering Press.
1242:, central to the solution of many
1105:, as shown in the figure at left.
1033:
995:
951:
919:
871:is an interaction coefficient and
199:light beam makes the beam scatter.
25:
3428:
2528:
1529:is involved (Roqué et al. 2006).
1161:corresponds to a solution to the
3361:
3360:
1953:. Royal Society of London: 3087.
1795:Electrophoretic light scattering
1727:Electrophoretic light scattering
1465:. Inelastic scattering includes
1381:
1140:Elastic and inelastic scattering
617:
464:
436:. The phenomenon stems from the
47:
2439:
2420:
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2313:
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2239:
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2181:
2046:"Recent Work with Röntgen Rays"
1972:. Royal Society of London: 770.
1392:needs additional citations for
1349:, the relevant equation is the
878:The above ordinary first-order
628:needs additional citations for
3310:Relativistic quantum mechanics
2156:
2127:
2108:
2086:
2056:(1383): 613â616. 30 Apr 1896.
2038:
2022:Merritt, Ernest (5 Oct 1898).
2015:
1976:
1957:
1938:
1895:Scattering from rough surfaces
1284:partial differential equations
1231:in quantum mechanics, while a
1001:
989:
756:partial differential equations
421:Single and multiple scattering
386:, introduced and developed by
13:
1:
3288:Quantum statistical mechanics
3065:Quantum differential calculus
2987:Delayed-choice quantum eraser
2770:Symmetry in quantum mechanics
2248:Multiple Scattering in Solids
2219:Multiple Scattering in Solids
2197:European Southern Observatory
1931:
1815:Light scattering by particles
1805:HaagâRuelle scattering theory
1508:. The apparent blue color of
1322:, the equations are those of
1148:The example of scattering in
777:Attenuation due to scattering
95:Light scattering by particles
2550:Neutron and X-Ray Scattering
1760:Characteristic mode analysis
1740:Attenuation#Light scattering
1306:). In the case of classical
1240:inverse scattering transform
1126:mass attenuation coefficient
262:particle-particle collisions
7:
3090:Quantum stochastic calculus
3080:Quantum measurement problem
3002:MachâZehnder interferometer
1732:
1355:Lippmann-Schwinger equation
882:has solutions of the form:
32:Scattering (disambiguation)
27:Range of physical processes
10:
3433:
2569:Optical Society of America
2560:Scattering and diffraction
1966:Philosophical Transactions
1947:Philosophical Transactions
1721:involves the migration of
771:inverse scattering problem
360:computer-generated imagery
335:in organisms, and textile
264:between molecules, atoms,
29:
3356:
3318:
3270:
3150:Quantum complexity theory
3128:Quantum cellular automata
3103:
3035:
2969:
2882:
2846:
2833:Path integral formulation
2800:
2665:
2622:
1527:surface plasmon resonance
1178:deep inelastic scattering
767:direct scattering problem
608:
3217:Quantum machine learning
3197:Quantum key distribution
3187:Quantum image processing
3177:Quantum error correction
3027:Wheeler's delayed choice
2275:Stover, John C. (1995).
2163:Nachtmann, Otto (1990).
1790:Dynamic Light Scattering
1134:inelastic mean free path
403:attenuation coefficients
399:scattering cross section
3133:Quantum finite automata
2193:ESO Picture of the Week
1900:Scintillation (physics)
1324:Quantum electrodynamics
1244:exactly solvable models
1118:attenuation coefficient
572:Coherent backscattering
3237:Quantum neural network
2545:Neutron Scattering Web
2098:Philosophical Magazine
1983:Tyndall, John (1874).
1885:Small-angle scattering
1708:finite-element methods
1577:
1447:
1328:Quantum chromodynamics
1271:
1184:Mathematical framework
1114:absorption coefficient
1054:
858:
804:
786:
550:passing through thick
457:
388:John Archibald Wheeler
200:
3262:Quantum teleportation
2790:Waveâparticle duality
2319:Ludwig Reimer (1997)
2299:R. F. Egerton (1996)
1925:X-ray crystallography
1880:Rutherford scattering
1780:Diffuse sky radiation
1775:Deep scattering layer
1598:, these domains are:
1578:
1536:which is defined as:
1451:Electromagnetic waves
1438:
1363:partial wave analysis
1336:fundamental particles
1257:
1194:differential equation
1128:(e.g. in cm/gram) or
1055:
880:differential equation
859:
805:
784:
739:(or angle change) of
737:Rutherford scattering
603:waveâparticle duality
428:
274:particle accelerators
191:
150:X-ray crystallography
3293:Quantum field theory
3222:Quantum metamaterial
3167:Quantum cryptography
2897:Consistent histories
1890:Scattering amplitude
1835:Molecular scattering
1755:Brillouin scattering
1540:
1467:Brillouin scattering
1401:improve this article
1351:Schrödinger equation
1276:mathematical physics
1163:Schrödinger equation
889:
817:
794:
752:inelastic scattering
637:improve this article
574:, an enhancement of
488:improve this section
30:For other uses, see
3278:Quantum fluctuation
3247:Quantum programming
3207:Quantum logic gates
3192:Quantum information
3172:Quantum electronics
2647:Classical mechanics
2518:Malvern Panalytical
2466:1998Natur.396...28P
2062:1896Natur..53..613.
2001:1874RSPT..164..183T
1870:Rayleigh scattering
1712:Maxwell's equations
1630:Rayleigh scattering
1607:Rayleigh scattering
1515:Rayleigh scattering
1459:Rayleigh scattering
1268:index of refraction
1250:Theoretical physics
1233:continuous spectrum
1074: −
560:multiple scattering
532:multiple scattering
446:interplanetary dust
352:semiconductor wafer
323:solids, defects in
227:diffuse reflections
3392:Physical phenomena
3331:in popular culture
3113:Quantum algorithms
2961:Von NeumannâWigner
2941:Objective collapse
2652:Old quantum theory
2514:"Light Scattering"
2446:Prum, Richard O.;
1915:Thomson scattering
1860:Powder diffraction
1845:Neutron scattering
1770:Coulomb scattering
1765:Compton scattering
1573:
1517:), the human blue
1506:elastic scattering
1479:Compton scattering
1448:
1367:Born approximation
1272:
1050:
854:
800:
787:
760:boundary condition
458:
448:spread throughout
348:medical ultrasound
293:scattering centers
278:neutron scattering
201:
110:Powder diffraction
3374:
3373:
3348:Quantum mysticism
3326:Schrödinger's cat
3257:Quantum simulator
3227:Quantum metrology
3155:Quantum computing
3118:Quantum amplifier
3095:Quantum spacetime
3060:Quantum cosmology
3050:Quantum chemistry
2765:Scattering theory
2713:Zero-point energy
2708:Degenerate levels
2616:Quantum mechanics
2448:Rodolfo H. Torres
2408:(16): 3813â3824.
2384:978-0-471-29340-8
2355:978-3-540-62838-5
2286:978-0-8194-1934-7
2261:978-0-387-98853-5
2232:978-0-387-98853-5
2149:978-3-540-62838-5
2028:Electrical Review
1750:Bragg diffraction
1559:
1433:
1432:
1425:
1359:Faddeev equations
1347:quantum chemistry
1345:, which includes
1343:quantum mechanics
1280:scattering theory
1225:discrete spectrum
1167:central potential
1150:quantum chemistry
1043:
961:
838:
803:{\displaystyle I}
713:
712:
705:
687:
580:weak localization
528:single scattering
524:
523:
516:
392:Werner Heisenberg
341:scattering theory
329:surface roughness
254:Ernest Rutherford
222:law of reflection
186:
185:
70:Bragg diffraction
18:Scatter (physics)
16:(Redirected from
3424:
3407:Particle physics
3364:
3363:
3075:Quantum geometry
3070:Quantum dynamics
2927:Superdeterminism
2823:Matrix mechanics
2678:Braâket notation
2609:
2602:
2595:
2586:
2585:
2522:
2521:
2510:
2504:
2503:
2500:Wyatt Technology
2492:
2486:
2485:
2443:
2437:
2424:
2418:
2417:
2395:
2389:
2388:
2366:
2360:
2359:
2337:
2331:
2317:
2311:
2297:
2291:
2290:
2272:
2266:
2265:
2243:
2237:
2236:
2214:
2208:
2207:
2205:
2203:
2185:
2179:
2178:
2160:
2154:
2153:
2131:
2125:
2112:
2106:
2105:
2090:
2084:
2083:
2073:
2071:10.1038/053613a0
2042:
2036:
2035:
2019:
2013:
2012:
1980:
1974:
1973:
1961:
1955:
1954:
1942:
1865:Raman scattering
1855:Photon diffusion
1701:geometric optics
1582:
1580:
1579:
1574:
1566:
1561:
1560:
1557:
1471:Raman scattering
1428:
1421:
1417:
1414:
1408:
1385:
1377:
1373:Electromagnetics
1320:particle physics
1302:, coming from a
1130:area per nucleon
1059:
1057:
1056:
1051:
1046:
1045:
1044:
1039:
1028:
1018:
1017:
1005:
1004:
977:
976:
964:
963:
962:
957:
949:
939:
938:
926:
925:
907:
906:
863:
861:
860:
855:
839:
837:
829:
821:
809:
807:
806:
801:
735:on a table, the
708:
701:
697:
694:
688:
686:
645:
621:
613:
558:, and the terms
536:diffusion theory
519:
512:
508:
505:
499:
468:
460:
380:particle physics
364:particle physics
311:fluctuations in
282:nuclear reactors
242:William Herschel
178:
171:
164:
51:
37:
36:
21:
3432:
3431:
3427:
3426:
3425:
3423:
3422:
3421:
3402:Nuclear physics
3377:
3376:
3375:
3370:
3352:
3338:Wigner's friend
3314:
3305:Quantum gravity
3266:
3252:Quantum sensing
3232:Quantum network
3212:Quantum machine
3182:Quantum imaging
3145:Quantum circuit
3140:Quantum channel
3099:
3045:Quantum biology
3031:
3007:ElitzurâVaidman
2982:DavissonâGermer
2965:
2917:Hidden-variable
2907:de BroglieâBohm
2884:Interpretations
2878:
2842:
2796:
2683:Complementarity
2661:
2618:
2613:
2531:
2526:
2525:
2512:
2511:
2507:
2494:
2493:
2489:
2460:(6706): 28â29.
2444:
2440:
2425:
2421:
2396:
2392:
2385:
2367:
2363:
2356:
2338:
2334:
2318:
2314:
2298:
2294:
2287:
2273:
2269:
2262:
2244:
2240:
2233:
2215:
2211:
2201:
2199:
2187:
2186:
2182:
2175:
2161:
2157:
2150:
2132:
2128:
2113:
2109:
2091:
2087:
2044:
2043:
2039:
2020:
2016:
1981:
1977:
1962:
1958:
1943:
1939:
1934:
1929:
1840:Mott scattering
1735:
1719:Electrophoresis
1589:
1562:
1556:
1552:
1541:
1538:
1537:
1477:scattering and
1441:Feynman diagram
1429:
1418:
1412:
1409:
1398:
1386:
1375:
1318:is studied. In
1308:electrodynamics
1252:
1186:
1142:
1080:
1069:
1029:
1027:
1023:
1019:
1013:
1009:
982:
978:
972:
968:
950:
948:
944:
940:
934:
930:
912:
908:
902:
898:
890:
887:
886:
830:
822:
820:
818:
815:
814:
795:
792:
791:
779:
741:alpha particles
709:
698:
692:
689:
646:
644:
634:
622:
611:
520:
509:
503:
500:
485:
469:
423:
372:nuclear physics
325:monocrystalline
321:polycrystalline
182:
62:
61:
55:Feynman diagram
35:
28:
23:
22:
15:
12:
11:
5:
3430:
3420:
3419:
3414:
3409:
3404:
3399:
3397:Atomic physics
3394:
3389:
3372:
3371:
3369:
3368:
3357:
3354:
3353:
3351:
3350:
3345:
3340:
3335:
3334:
3333:
3322:
3320:
3316:
3315:
3313:
3312:
3307:
3302:
3301:
3300:
3290:
3285:
3283:Casimir effect
3280:
3274:
3272:
3268:
3267:
3265:
3264:
3259:
3254:
3249:
3244:
3242:Quantum optics
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3164:
3163:
3162:
3152:
3147:
3142:
3137:
3136:
3135:
3125:
3120:
3115:
3109:
3107:
3101:
3100:
3098:
3097:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3041:
3039:
3033:
3032:
3030:
3029:
3024:
3019:
3017:Quantum eraser
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2973:
2971:
2967:
2966:
2964:
2963:
2958:
2953:
2948:
2943:
2938:
2933:
2932:
2931:
2930:
2929:
2914:
2909:
2904:
2899:
2894:
2888:
2886:
2880:
2879:
2877:
2876:
2871:
2866:
2861:
2856:
2850:
2848:
2844:
2843:
2841:
2840:
2835:
2830:
2825:
2820:
2815:
2810:
2804:
2802:
2798:
2797:
2795:
2794:
2793:
2792:
2787:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2716:
2715:
2710:
2705:
2700:
2690:
2688:Density matrix
2685:
2680:
2675:
2669:
2667:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2643:
2642:
2632:
2626:
2624:
2620:
2619:
2612:
2611:
2604:
2597:
2589:
2583:
2582:
2577:
2572:
2562:
2557:
2552:
2547:
2542:
2537:
2530:
2529:External links
2527:
2524:
2523:
2505:
2487:
2438:
2419:
2390:
2383:
2361:
2354:
2332:
2312:
2292:
2285:
2267:
2260:
2238:
2231:
2209:
2180:
2173:
2155:
2148:
2126:
2107:
2094:Rutherford, E.
2085:
2037:
2014:
1975:
1956:
1936:
1935:
1933:
1930:
1928:
1927:
1922:
1917:
1912:
1910:Tyndall effect
1907:
1902:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1825:Mie scattering
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1785:Doppler effect
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1745:Backscattering
1742:
1736:
1734:
1731:
1723:macromolecules
1670:Mie scattering
1627:
1626:
1620:
1617:Mie scattering
1610:
1587:
1572:
1569:
1565:
1555:
1551:
1548:
1545:
1463:Mie scattering
1431:
1430:
1389:
1387:
1380:
1374:
1371:
1332:Standard Model
1251:
1248:
1227:correspond to
1185:
1182:
1141:
1138:
1083:mean free path
1078:
1067:
1061:
1060:
1049:
1042:
1038:
1035:
1032:
1026:
1022:
1016:
1012:
1008:
1003:
1000:
997:
994:
991:
988:
985:
981:
975:
971:
967:
960:
956:
953:
947:
943:
937:
933:
929:
924:
921:
918:
915:
911:
905:
901:
897:
894:
865:
864:
851:
848:
845:
842:
836:
833:
828:
825:
799:
778:
775:
733:billiard balls
711:
710:
625:
623:
616:
610:
607:
576:backscattering
522:
521:
472:
470:
463:
430:Zodiacal light
422:
419:
415:mean free path
356:polymerization
197:LCD projectors
184:
183:
181:
180:
173:
166:
158:
155:
154:
153:
152:
147:
142:
137:
132:
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
77:
72:
64:
63:
53:
52:
44:
43:
26:
9:
6:
4:
3:
2:
3429:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3384:
3382:
3367:
3359:
3358:
3355:
3349:
3346:
3344:
3341:
3339:
3336:
3332:
3329:
3328:
3327:
3324:
3323:
3321:
3317:
3311:
3308:
3306:
3303:
3299:
3296:
3295:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3275:
3273:
3269:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3202:Quantum logic
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3161:
3158:
3157:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3134:
3131:
3130:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3110:
3108:
3106:
3102:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3055:Quantum chaos
3053:
3051:
3048:
3046:
3043:
3042:
3040:
3038:
3034:
3028:
3025:
3023:
3022:SternâGerlach
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2972:
2968:
2962:
2959:
2957:
2956:Transactional
2954:
2952:
2949:
2947:
2946:Quantum logic
2944:
2942:
2939:
2937:
2934:
2928:
2925:
2924:
2923:
2920:
2919:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2889:
2887:
2885:
2881:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2851:
2849:
2845:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2805:
2803:
2799:
2791:
2788:
2786:
2783:
2782:
2781:
2780:Wave function
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2755:Superposition
2753:
2751:
2750:Quantum state
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2714:
2711:
2709:
2706:
2704:
2703:Excited state
2701:
2699:
2696:
2695:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2668:
2664:
2658:
2655:
2653:
2650:
2648:
2645:
2641:
2638:
2637:
2636:
2633:
2631:
2628:
2627:
2625:
2621:
2617:
2610:
2605:
2603:
2598:
2596:
2591:
2590:
2587:
2581:
2578:
2576:
2573:
2570:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2519:
2515:
2509:
2501:
2497:
2491:
2483:
2479:
2475:
2474:10.1038/23838
2471:
2467:
2463:
2459:
2455:
2454:
2449:
2442:
2436:
2435:0-471-82857-2
2432:
2428:
2423:
2415:
2411:
2407:
2403:
2402:
2394:
2386:
2380:
2376:
2372:
2365:
2357:
2351:
2347:
2343:
2336:
2330:
2329:3-540-62568-2
2326:
2322:
2316:
2310:
2309:0-306-45223-5
2306:
2302:
2296:
2288:
2282:
2278:
2271:
2263:
2257:
2253:
2249:
2242:
2234:
2228:
2224:
2220:
2213:
2198:
2194:
2190:
2184:
2176:
2174:3-540-50496-6
2170:
2166:
2159:
2151:
2145:
2141:
2137:
2130:
2124:
2123:0-471-82857-2
2120:
2116:
2111:
2103:
2099:
2095:
2089:
2081:
2077:
2072:
2067:
2063:
2059:
2055:
2051:
2047:
2041:
2033:
2029:
2025:
2018:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1979:
1971:
1967:
1960:
1952:
1948:
1941:
1937:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
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1836:
1833:
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1826:
1823:
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1806:
1803:
1801:
1798:
1796:
1793:
1791:
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1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
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1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1737:
1730:
1728:
1724:
1720:
1716:
1713:
1709:
1704:
1702:
1697:
1695:
1691:
1687:
1686:Doppler shift
1681:
1679:
1675:
1671:
1667:
1662:
1660:
1657:, angle, and
1656:
1652:
1648:
1644:
1640:
1636:
1635:Lord Rayleigh
1631:
1624:
1621:
1618:
1614:
1611:
1608:
1604:
1601:
1600:
1599:
1597:
1593:
1586:
1570:
1567:
1563:
1553:
1549:
1546:
1543:
1535:
1530:
1528:
1524:
1523:nanoparticles
1520:
1516:
1511:
1507:
1502:
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1496:
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1487:
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1476:
1472:
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1427:
1424:
1416:
1406:
1402:
1396:
1395:
1390:This section
1388:
1384:
1379:
1378:
1370:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1339:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1292:wave equation
1289:
1285:
1281:
1277:
1269:
1265:
1261:
1256:
1247:
1245:
1241:
1236:
1234:
1230:
1226:
1222:
1218:
1217:Hilbert space
1214:
1210:
1206:
1201:
1199:
1195:
1191:
1181:
1179:
1174:
1172:
1168:
1164:
1160:
1159:hydrogen atom
1156:
1151:
1146:
1137:
1135:
1131:
1127:
1123:
1119:
1115:
1111:
1106:
1104:
1103:ρ/τ
1100:
1096:
1092:
1088:
1087:cross-section
1084:
1077:
1073:
1066:
1047:
1040:
1036:
1030:
1024:
1020:
1014:
1010:
1006:
998:
992:
986:
983:
979:
973:
969:
965:
958:
954:
945:
941:
935:
931:
927:
922:
916:
913:
909:
903:
899:
895:
892:
885:
884:
883:
881:
876:
874:
870:
849:
846:
843:
840:
834:
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810:), i.e. that
797:
783:
774:
772:
768:
763:
761:
757:
753:
749:
746:
742:
738:
734:
730:
726:
722:
718:
707:
704:
696:
685:
682:
678:
675:
671:
668:
664:
661:
657:
654: â
653:
649:
648:Find sources:
642:
638:
632:
631:
626:This section
624:
620:
615:
614:
606:
604:
599:
597:
591:
589:
583:
581:
577:
573:
569:
565:
561:
557:
553:
549:
543:
539:
537:
533:
529:
518:
515:
507:
497:
493:
489:
483:
482:
478:
473:This section
471:
467:
462:
461:
455:
451:
447:
443:
439:
435:
431:
427:
418:
416:
412:
408:
404:
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389:
385:
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377:
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369:
365:
361:
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344:
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334:
330:
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322:
318:
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279:
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258:
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148:
146:
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138:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
90:Kikuchi lines
88:
86:
85:Dynamic light
83:
81:
78:
76:
73:
71:
68:
67:
66:
65:
60:
56:
50:
46:
45:
42:
39:
38:
33:
19:
3412:Radar theory
3085:Quantum mind
2997:FranckâHertz
2859:KleinâGordon
2808:Formulations
2801:Formulations
2730:Interference
2720:Entanglement
2698:Ground state
2693:Energy level
2666:Fundamentals
2630:Introduction
2517:
2508:
2499:
2490:
2457:
2451:
2441:
2422:
2405:
2399:
2393:
2370:
2364:
2341:
2335:
2320:
2315:
2300:
2295:
2276:
2270:
2247:
2241:
2218:
2212:
2200:. Retrieved
2192:
2183:
2164:
2158:
2135:
2129:
2110:
2101:
2097:
2088:
2053:
2049:
2040:
2031:
2027:
2017:
1992:
1988:
1978:
1969:
1965:
1959:
1950:
1946:
1940:
1810:Kikuchi line
1717:
1710:which solve
1705:
1698:
1682:
1663:
1655:polarization
1646:
1642:
1628:
1622:
1612:
1602:
1595:
1591:
1584:
1533:
1531:
1503:
1485:
1483:
1473:, inelastic
1449:
1419:
1413:January 2020
1410:
1399:Please help
1394:verification
1391:
1340:
1279:
1273:
1237:
1229:bound states
1202:
1187:
1175:
1147:
1143:
1129:
1107:
1102:
1099:ησ
1098:
1094:
1090:
1075:
1071:
1064:
1062:
877:
872:
868:
866:
788:
766:
764:
714:
699:
690:
680:
673:
666:
659:
652:"Scattering"
647:
635:Please help
630:verification
627:
600:
592:
584:
567:
563:
559:
544:
540:
531:
527:
525:
510:
504:January 2017
501:
486:Please help
474:
454:Solar System
396:
376:astrophysics
354:inspection,
345:
317:crystallites
292:
288:
286:
259:
250:cathode rays
246:John Tyndall
238:Isaac Newton
231:
225:
204:
203:In physics,
202:
40:
3343:EPR paradox
3123:Quantum bus
2992:Double-slit
2970:Experiments
2936:Many-worlds
2874:Schrödinger
2838:Phase space
2828:Schrödinger
2818:Interaction
2775:Uncertainty
2745:Nonlocality
2740:Measurement
2735:Decoherence
2725:Hamiltonian
1920:Wolf effect
1341:In regular
1316:radio waves
1296:sound waves
1190:mathematics
1155:bound state
693:August 2023
145:Wolf effect
130:Small-angle
3387:Scattering
3381:Categories
3271:Extensions
3105:Technology
2951:Relational
2902:Copenhagen
2813:Heisenberg
2760:Tunnelling
2623:Background
2202:2 December
2034:(14): 217.
1932:References
1830:Mie theory
1800:Extinction
1678:ellipsoids
1666:Gustav Mie
1651:atmosphere
1639:wavelength
1365:, and the
1264:plane wave
1176:The term "
727:to form a
725:rain drops
663:newspapers
548:light beam
438:scattering
411:S-matrices
289:scatterers
270:cosmic ray
217:trajectory
205:scattering
193:Wine glass
125:Rutherford
41:Scattering
2977:Bell test
2847:Equations
2673:Born rule
1820:Linewidth
1674:spheroids
1659:coherence
1568:λ
1550:π
1544:α
1304:submarine
1300:sea water
1288:acoustics
1260:real part
1258:Top: the
1205:manifolds
1093:= 1/
1041:τ
1034:Δ
1031:ρ
1025:−
996:Δ
993:η
987:σ
984:−
959:λ
952:Δ
946:−
920:Δ
914:−
844:−
721:particles
568:diffusers
564:diffusion
556:diffusion
475:does not
450:the plane
434:night sky
297:particles
266:electrons
75:Brillouin
3366:Category
3160:Timeline
2912:Ensemble
2892:Bayesian
2785:Collapse
2657:Glossary
2640:Timeline
2427:Seinfeld
2346:Springer
2252:Springer
2223:Springer
2140:Springer
2115:Seinfeld
1905:S-Matrix
1733:See also
1357:and the
1330:and the
1221:S matrix
1213:operator
1209:spectrum
596:speckles
442:sunlight
409:(BSDF),
384:S-Matrix
327:solids,
305:droplets
233:specular
120:Rayleigh
3319:Related
3298:History
3037:Science
2869:Rydberg
2635:History
2482:4393904
2462:Bibcode
2080:4023635
2058:Bibcode
1997:Bibcode
1995:: 221.
1171:ionized
1110:opacity
1101:=
1097:=
729:rainbow
677:scholar
496:removed
481:sources
452:of the
309:density
301:bubbles
280:inside
140:Thomson
135:Tyndall
105:Neutron
80:Compton
3012:Popper
2571:, 1997
2480:
2453:Nature
2433:
2381:
2352:
2327:
2307:
2283:
2258:
2229:
2171:
2146:
2121:
2078:
2050:Nature
2009:109101
2007:
1583:where
1495:lustre
1445:photon
1211:of an
1116:, and
1095:λ
1063:where
867:where
748:nuclei
679:
672:
665:
658:
650:
609:Theory
413:, and
405:, the
337:fibers
313:fluids
59:photon
2922:Local
2864:Pauli
2854:Dirac
2478:S2CID
2375:Wiley
2104:: 21.
2076:S2CID
2005:JSTOR
1694:radar
1690:lidar
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1605:âȘ 1:
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1486:white
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1262:of a
1122:barns
717:waves
684:JSTOR
670:books
588:radar
401:(Ï),
378:. In
333:cells
213:sound
209:light
115:Raman
2431:ISBN
2379:ISBN
2350:ISBN
2325:ISBN
2305:ISBN
2281:ISBN
2256:ISBN
2227:ISBN
2204:2013
2169:ISBN
2144:ISBN
2119:ISBN
1692:and
1676:and
1519:iris
1493:(or
1461:and
1198:time
765:The
745:gold
719:and
656:news
562:and
479:any
477:cite
390:and
374:and
2470:doi
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2410:doi
2066:doi
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