Scattering

Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

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 straight trajectory by one or more localized non-uniformities in the medium through which it passes. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections that undergo scattering are often called diffuse reflections and unscattered reflections are called specular (mirror-like) reflections.

The types of non-uniformities that 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, defects in crystalline 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.

Elastic and inelastic scattering

In physical descriptions of scattering, physicists commonly distinguish between two broad types, elastic and inelastic. Elastic scattering involves no (or a very small) loss or gain of energy by the radiation, whereas inelastic scattering does involve some change in the energy of the radiation. These terms derive from the kinetic theory of the collision of solid bodies, which can often be used to model scattering processes.

If the radiation is substantially or completely extinguished by the interaction (losing a significant proportion of its energy), the process is known as absorption. In some contexts, absorption is considered to be merely an extreme form of inelastic scattering. Generally speaking, in classical physics absorption and scattering tend to be treated as different phenomena, while in quantum physics absorption is treated as a form of scattering via the S-matrix. To be precise, absorption cannot occur without some degree of scattering, and scattering is rarely completely elastic, but on a macroscopic scale it is common for "absorption" and "scattering" to take place without any appreciable contribution from the other process, so treating them separately is often convenient.

Single and multiple scattering

When radiation is only scattered by one localized scattering center, this is called single scattering. It is very common that scattering centers are grouped together, and in those cases the radiation may scatter many times, which is known as multiple scattering. The main difference between the effects of single and multiple scattering is that single scattering can usually be treated as a random phenomenon and multiple scattering is usually more deterministic. Since the location of a single often microscopic 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 an atomic nucleus. In that case, the atom's exact position relative to the path of the electron is unknown and would be immeasurable, so the exact direction of the electron after the collision is unknown, plus the quantum-mechanical nature of this particular interaction also makes the interaction random. Single scattering is therefore often described by probability distributions.

With multiple scattering, the randomness of the interaction tends to be averaged out by the large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity as the radiation is spread out. This is exemplified by 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 thus known as diffusers.

Not all single scattering is random, however, as a well-controlled laser beam can be exactly positioned to scatter off a microscopic particle with a deterministic outcome. Such situations are encountered in radar scattering as well, where the targets tend to be macroscopic objects such as people or aircraft.

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 speckles. Speckle also occurs if multiple parts of a coherent wave scatter from different centers. In certain rare circumstances, multiple scattering may only involve 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.

The description of scattering and the distinction between single and multiple scattering are often highly involved with wave-particle duality.

Major research problems in scattering often involve predicting how various systems will scatter radiation, which can almost always be solved given sufficient computing power and knowledge of the system. A widely studied but more difficult challenge is the inverse scattering problem, in which the goal is to observe scattered radiation and use that observation to determine properties of either the scatterer or the radiation before scattering. In general, the inverse is not unique; several different types of scattering centers can usually give rise to the same pattern of scattered radiation, so the problem is not solvable in the general case. Fortunately, there are ways to extract some useful, albeit incomplete, information about the scatterer, and these techniques are widely used for sensing and metrology applications (Colton & Kress 1998).

Some areas where scattering and scattering theory are significant include radar sensing, medical ultrasound, semiconductor wafer inspection, polymerization process monitoring, acoustic tiling, free-space communications, and computer-generated imagery.

Electromagnetic scattering

File:Electron-scattering.png

A Feynman diagram of scattering between two electrons by emission of a virtual photon.

Electromagnetic (EM) waves are one of the best known and most commonly encountered forms of radiation that undergo scattering. Scattering of light and radio waves (especially in radar) 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 Rayleigh scattering and Mie scattering. Inelastic EM scattering effects include 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 almost completely to the scattering of light by the surface of the object. The absence of surface scattering leads to a shiny or glossy appearance. Light scattering can also give color to some objects, usually shades of blue (as with the sky, the human iris, and the feathers of some birds (Prum et al. 1998), but resonant light scattering in nanoparticles can produce different highly saturated and vibrant hues, especially when surface plasmon resonance is involved (Roqué et al. 2006).

Rayleigh scattering is a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive index, such as a particle, bubble, droplet, or even a density fluctuation. This effect was first modeled successfully by Lord Rayleigh, from whom it gets its name. In order for Rayleigh's model to apply, the sphere must be much smaller in diameter than the wavelength (λ) 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/λ ⁴ relation. Along with absorption, such scattering is a major cause of the attenuation of radiation by the atmosphere. 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 polarization, angle, and coherence.

For larger diameters, the problem of electromagnetic scattering by spheres was first solved by Gustav Mie, and scattering by spheres larger than the Rayleigh range is therefore usually known as Mie scattering. 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, spheroids and ellipsoids. Closed-form solutions for scattering by certain other simple shapes exist, but no general closed-form solution is known for arbitrary shapes.

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 Doppler shift, which can be detected and used to measure the velocity of the scattering center/s in forms of techniques such as LIDAR and radar. This shift involves a slight change in energy.

At values of the ratio of particle diameter to wavelength more than about 10, the laws of geometric optics are mostly sufficient to describe the interaction of light with the particle, and at this point the interaction is not usually described as scattering.

For modeling of scattering in cases where the Rayleigh and Mie models do not apply such as irregularly shaped particles, there are many numerical methods that can be used. The most common are finite-element methods which solve Maxwell's equations 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 structures, these models usually require substantial execution times on a computer.

Another special type of EM scattering is coherent backscattering. This is a relatively obscure phenomenon that occurs when coherent radiation (such as a laser beam) propagates through a medium which has a large number of scattering centers, so that the waves are scattered many times while traveling through it. A thick cloud is a typical example of this sort of multiple-scattering medium. The effect produces a very large peak in the scattering intensity in the direction from the which the wave travels—effectively, the light scatters preferentially back the way it came. For incoherent radiation, the scattering typically reaches a local maximum in the backward direction, but the coherent backscatter peak is two times higher than the level would have been if the light were incoherent. It is very difficult to detect and measure for two reasons. The first is fairly obvious, that it is difficult to measure the direct backscatter without blocking the beam, but there are methods for overcoming this problem. The second is that the peak is usually extremely sharp around the backward direction, so that a very high level of angular resolution is needed for the detector to see the peak without averaging its intensity out over the surrounding angles where the intensity can undergo large dips. At angles other than the backscatter direction, the light intensity is subject to numerous essentially random fluctuations called speckles.

This is one of the most robust interference phenomena that survives multiple scattering, and it is regarded as an aspect of a quantum mechanical phenomenon known as weak localization (Akkermans et al. 1986). In weak localization, interference of the direct and reverse paths leads to a net reduction of light transport in the forward direction. This phenomenon is typical of any coherent wave which is multiple scattered. It is typically discussed for light waves, for which it is similar to the weak localization phenomenon for electrons in disordered (semi)conductors and often seen as the precursor to Anderson (or strong) localization of light. Weak localization of light can be detected since it is manifested as an enhancement of light intensity in the backscattering direction. This substantial enhancement is called the cone of coherent backscattering .

Coherent backscattering has its origin in the interference between direct and reverse paths in the backscattering direction. When a multiply scattering medium is illuminated by a laser beam, the scattered intensity results from the interference between the amplitudes associated with the various scattering paths; for a disordered medium, the interference terms are washed out when averaged over many sample configurations, except in a narrow angular range around exact backscattering where the average intensity is enhanced. This phenomenon, is the result of many sinusoidal two-waves interference patterns which add up. The cone is the Fourier transform of the spatial distribution of the intensity of the scattered light on the sample surface, when the latter is illuminated by a point-like source. The enhanced backscattering relies on the constructive interference between reverse paths. One can make an analogy with a Young's interference experiment, where two diffracting slits would be positioned in place of the "input" and "output" scatterers.

Scattering in particle physics

In particle physics, scattering refers to deflection of subatomic particles, a process central to many experiments. In scattering experiments, a target of some material is bombarded with a beam of particles (typically electrons, protons, or neutrons) and the number of particles emerging in various directions is measured. This distribution reveals information about the interaction that takes place between the target and the scattered particle.

Depending on the type of interaction between beam-particle and target, the scattering process is called elastic, inelastic or deeply inelastic. In an elastic scattering process the incident and target particles are left intact and only their momenta may be changed. In an inelastic scattering process the target particle is excited. For example if a nucleus is bombarded by neutrons, it may be excited to some nuclear resonance. In deep inelastic scattering the target (and sometimes the incident particle) is destroyed and completely new particles may be created.

A famous scattering experiment of alpha particles off gold nuclei performed by Ernest Rutherford revealed the basic structure of the atom - a tiny nucleus surrounded by electrons. See Rutherford scattering. Scattering has also been done off of nucleons and quarks.

Mathematically, scientists describe scattering by an impact parameter (which describes how close the incident particle would come to the target if it moved in a straight line) and an angle of deflection (which describes the angle at which the particle emerges relative to its original direction). The distribution of deflection angles is described by a function known as the differential cross section, which (roughly) relates a direction in space in which some particles emerge to the amount of the incoming beam (in area) those particles came from. In more complicated cases of scattering, such as deep inelastic scattering of electrons and protons, so-called form factors have to be multiplied to the scattering formulae, describing the internal structure of the proton.

The abstract mathematics of scattering is developed as scattering theory.

See also S-matrix.

See also

• Brillouin scattering
• Compton scattering
• CREIL (Coherent Raman Effect on Incoherent Light)
• Dynamic Light Scattering
• Neutron scattering
• Light scattering by particles
• Mie theory
• Raman scattering
• Rayleigh scattering
• Rutherford scattering
• Small-angle scattering
• Tyndall effect
• Thomson scattering
• Wolf effect

External links

• Research group on light scattering and diffusion in complex systems
• Neutron Scattering Web

References

• Akkermans, E. (1986). "Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape". Phys. Rev. Lett. 56 (14): 1471–1474.
• Bohren, Craig F. (1983). Absorption and Scattering of Light by Small Particles. Wiley. ISBN 0-471-29340-7.
• Colton, David (1998). Inverse Acoustic and Electromagnetic Scattering Theory. Springer. ISBN 3-540-62838-X.
• Gonis, Antonios (1999). Multiple Scattering in Solids. Springer. ISBN 0-387-98853-X.
• Prum, Richard O. (1998). "Coherent light scattering by blue feather barbs". Nature. 396 (6706): 28–29.
• Roqué, Josep (2006). "Copper and silver nanocrystals in lustre lead glazes: development and optical properties". J. Eur. Ceramic Society. 26 (16): 3813–3824.
• Stover, John C. (1995). Optical Scattering: Measurement and Analysis. SPIE Optical Engineering Press. ISBN 0-8194-1934-6.

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