• scattering
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Aspect-oriented software development Problems caused by scattering and
tangling
1
Scattering and tangling of behavior are
the symptoms that the implementation
of a concern is not well modularized. A
concern that is not modularized does not
exhibit a well-defined interface. The
interactions between the
implementation of the concern and the
modules of the system are not explicitly
declared. They are encoded implicitly
through the dependencies and
interactions between fragments of code
that implement the concern and the
implementation of other modules.
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Aspect-oriented software development Problems caused by scattering and
tangling
1
The lack of interfaces between the
implementation of crosscutting
concerns and the implementation of
the modules of the system impedes
the development, the evolution and
the maintenance of the system.
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Rayleigh scattering
1
Rayleigh scattering results from the electric
polarizability of the particles
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Rayleigh scattering
Rayleigh scattering of sunlight in the
atmosphere causes diffuse sky radiation,
which is the reason for the blue color of
the sky and the yellow tone of the sun
itself.
1
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Rayleigh scattering
Scattering by particles similar to or larger
than the wavelength of light is typically
treated by the Mie theory, the discrete dipole
approximation and other computational
techniques. Rayleigh scattering applies to
particles that are small with respect to
wavelengths of light, and that are optically
"soft" (i.e. with a refractive index close to 1).
On the other hand, Anomalous Diffraction
Theory applies to optically soft but larger
particles.
1
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Rayleigh scattering - Small size parameter approximation
1
Rayleigh scattering can be defined as
scattering in the small size parameter
regime x ≪ 1. Scattering from larger
spherical particles is explained by the
Mie theory for an arbitrary size
parameter x. For small x the Mie
theory reduces to the Rayleigh
approximation.
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Rayleigh scattering - Small size parameter approximation
The amount of Rayleigh scattering that
occurs for a beam of light depends upon
the size of the particles and the
wavelength of the light. Specifically, the
intensity of the scattered light varies as the
sixth power of the particle size, and varies
inversely with the fourth power of the
wavelength.
1
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Rayleigh scattering - Small size parameter approximation
1
The intensity I of light scattered by a single
small particle from a beam of unpolarized
light of wavelength λ and intensity I0 is
given by:
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Rayleigh scattering - Small size parameter approximation
The Rayleigh scattering coefficient for a
group of scattering particles is the number of
particles per unit volume N times the crosssection. As with all wave effects, for
incoherent scattering the scattered powers
add arithmetically, while for coherent
scattering, such as if the particles are very
near each other, the fields add arithmetically
and the sum must be squared to obtain the
total scattered power.
1
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Rayleigh scattering - From molecules
Rayleigh scattering also occurs from
individual molecules. Here the scattering is
due to the molecular polarizability α, which
describes how much the electrical charges
on the molecule will move in an electric
field. In this case, the Rayleigh scattering
intensity for a single particle is given by
1
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Rayleigh scattering - From molecules
1
The amount of Rayleigh scattering from a
single particle can also be expressed as a
cross section σ. For example, the major
constituent of the atmosphere, nitrogen,
has a Rayleigh cross section of 5.1×10−31
m2 at a wavelength of 532 nm (green
light). This means that at atmospheric
pressure, about a fraction 10−5 of light will
be scattered for every meter of travel.
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Rayleigh scattering - From molecules
1
The strong wavelength dependence of
the scattering (~λ−4) means that shorter
(blue) wavelengths are scattered more
strongly than longer (red) wavelengths.
This results in the indirect blue light
coming from all regions of the sky.
Rayleigh scattering is a good
approximation of the manner in which
light scattering occurs within various
media for which scattering particles
have a small size parameter.
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Rayleigh scattering - Reason for the blue color of the sky
1
However, the Sun, like any star, has its
own spectrum and so I0 in the
scattering formula above is not
constant but falls away in the violet
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Rayleigh scattering - Reason for the blue color of the sky
1
The reddening of sunlight is intensified
when the sun is near the horizon, because
the volume of air through which sunlight
must pass is significantly greater than
when the sun is high in the sky. The
Rayleigh scattering effect is thus
increased, removing virtually all blue light
from the direct path to the observer. The
remaining unscattered light is mostly of a
longer wavelength, and therefore appears
to be orange.
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Rayleigh scattering - Reason for the blue color of the sky
Some of the scattering can also be
from sulfate particles. For years after
large Plinian eruptions, the blue cast
of the sky is notably brightened by the
persistent sulfate load of the
stratospheric gases. Some works of the
artist J. M. W. Turner may owe their
vivid red colours to the eruption of
Mount Tambora in his lifetime.
1
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Rayleigh scattering - Reason for the blue color of the sky
1
In locations with little light pollution, the
moonlit night sky is also blue, because
moonlight is reflected sunlight, with a
slightly lower color temperature due to the
brownish color of the moon. The moonlit
sky is not perceived as blue, however,
because at low light levels human vision
comes mainly from rod cells that do not
produce any color perception (Purkinje
effect).
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Rayleigh scattering - In optical fibers
1
Rayleigh scattering is an important
component of the scattering of optical
signals in optical fibers. Silica fibers
are disordered materials, thus their
density varies on a microscopic scale.
The density fluctuations give rise to
energy loss due to the scattered light,
with the following coefficient:
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Rayleigh scattering - In optical fibers
1
where n is the refraction index, p is
the photoelastic coefficient of the
glass, k is the Boltzmann constant,
and β is the isothermal
compressibility. Tf is a fictive
temperature, representing the
temperature at which the density
fluctuations are "frozen" in the
material.
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Rayleigh scattering - In porous materials
λ−4 Rayleigh-type scattering can also be
exhibited by porous materials. An example is
the strong optical scattering by nanoporous
materials. The strong contrast in refractive
index between pores and solid parts of
sintered alumina results in very strong
scattering, with light completely changing
direction each 5 micrometers on average.
The λ−4-type scattering is caused by the
nanoporous structure (a narrow pore size
distribution around ~70 nm) obtained by
sintering monodispersive alumina powder.
1
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Rayleigh scattering - Further reading
1
Pedro Lilienfeld, "A Blue Sky History."
(2004). Optics and Photonics News. Vol.
15, Issue 6, pp. 32–39.
doi:10.1364/OPN.15.6.000032. Gives a
brief history of theories of why the sky
is blue leading up to Rayleigh's
discovery, and a brief description of
Rayleigh scattering.
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Library and information science - Scattering of the literature
1
Meho & Spurgin (2005) found that in a list
of 2,625 items published between 1982
and 2002 by 68 faculty members of 18
schools of library and information science,
only 10 databases provided significant
coverage of the LIS literature
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Library and information science - Scattering of the literature
1
"The study confirms earlier research
that LIS literature is highly scattered
and is not limited to standard LIS
databases
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Optical fiber - Light scattering
The propagation of light through the
core of an optical fiber is based on total
internal reflection of the lightwave.
Rough and irregular surfaces, even at
the molecular level, can cause light
rays to be reflected in random
directions. This is called diffuse
reflection or scattering, and it is
typically characterized by wide variety
of reflection angles.
1
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Optical fiber - Light scattering
Light scattering depends on the
wavelength of the light being scattered.
Thus, limits to spatial scales of visibility
arise, depending on the frequency of the
incident light-wave and the physical
dimension (or spatial scale) of the
scattering center, which is typically in
the form of some specific microstructural feature. Since visible light has
a wavelength of the order of one
micrometer (one millionth of a meter)
scattering centers will have dimensions
on a similar spatial scale.
1
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Optical fiber - Light scattering
1
It has recently been shown that when the
size of the scattering center (or grain
boundary) is reduced below the size of the
wavelength of the light being scattered,
the scattering no longer occurs to any
significant extent
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Optical fiber - Light scattering
1
Distributed both between and within these
domains are micro-structural defects that
provide the most ideal locations for light
scattering
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Optical fiber - Light scattering
At high optical powers, scattering can also be
caused by nonlinear optical processes in the fiber.
1
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Electrophoretic light scattering
1
'Electrophoretic light scattering' is
based on dynamic light scattering.
The frequency shift or phase
(waves)|phase shift of an incident
laser beam depends on the dispersed
particles mobility. In the case of
dynamic light scattering, Brownian
motion causes particle motion. In the
case of electrophoretic light
scattering, oscillating electric field
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Electrophoretic light scattering
This method is used for measuring
electrophoretic mobility and then
calculating zeta potential. Instruments to
apply the method are commercially
available from several manufacturers. The
last set of calculations requires
information on viscosity and dielectric
permittivity of the dispersion medium.
Appropriate electrophoresis theory is also
required. Sample dilution is often
necessary in order to eliminate particle
interactions.
1
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Electrophoretic light scattering - Instrumentation of Electrophretic light scattering
A laser beam passes through the
electrophoresis cell, irradiates the
particles dispersed in it, and is
scattered by the particles. The
scattered light is detected by a photomultiplier after passing through two
pinholes. There are two types of
optical systems: heterodyne and
fringe.
1
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Electrophoretic light scattering - Instrumentation of Electrophretic light scattering
Ware and Flygare (9) developed a
heterodyne-type ELS instrument, that
was the first instrument of this type. In a
fringe optics ELS instrument (10), a laser
beam is divided into two beams. Those
cross inside the electrophresis cell at a
fixed angle to produce a fringe pattern.
The scattered light from the particles,
which migrates inside the fringe, is
amplitude-modulated. The frequency
shifts from both types of optics obey the
same equations. The observed spectra
resemble each other.
1
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Electrophoretic light scattering - Instrumentation of Electrophretic light scattering
1
Oka et al. developed an ELS instrument of
heterodyne-type optics (11) that is now
available commercially. Its optics is shown
in Fig. 3.
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Electrophoretic light scattering - Instrumentation of Electrophretic light scattering
A modulator is a translationally moving
mirror. A frequency of reference light shifts
by the movement. Scattered light from
dispersed particle without electrophoretic
movement is line broadened. That with
electrophoretic movement is line
broadened and Doppler shifted. The
moving mirror moves to shorten optical
pass length, then the frequency shift
toward higher frequency.
1
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Electrophoretic light scattering - Heterodyne light scattering
The frequency of light scattered by
particles undergoing electrophoresis is
shifted by the amount of the Doppler
effect, \upsilon_D\, from that of the
incident light, :\upsilon\, .
1
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Electrophoretic light scattering - Heterodyne light scattering
1
The autocorrelation function of intensity of
the mixed light, g(\tau) \,, can be
approximately described by the following
damped cosine function [7].
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Electrophoretic light scattering - Heterodyne light scattering
: g(\tau)=A+B \exp(\Gamma\tau)\cos(2\pi\upsilon_o)+C \exp(-2\Gamma
\tau)\,\qquad (1)
1
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Electrophoretic light scattering - Heterodyne light scattering
1
Damping frequency \upsilon_o\, is
an observed frequency, and is the
frequency difference between
scattered and reference light.
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Electrophoretic light scattering - Heterodyne light scattering
1
where \upsilon_s\, is the frequency of
scattered light, \upsilon_r\, the
frequency of the reference light,
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Electrophoretic light scattering - Heterodyne light scattering
\upsilon_i\, the
frequency of incident
light (laser light),
1
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Electrophoretic light scattering - Heterodyne light scattering
1
and \upsilon_M\, the
modulation frequency.
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Electrophoretic light scattering - Heterodyne light scattering
The power spectrum of mixed light,
namely the Fourier transform of g(\tau) \,,
gives a couple of Lorenz functions at
\pm\Delta\upsilon \, having a half-width of
\Gamma/2\pi\, at the half maximum.
1
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Electrophoretic light scattering - Heterodyne light scattering
1
In addition to these two, the last term in equation (1)
gives another Lorenz function at
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Electrophoretic light scattering - Heterodyne light scattering
1
The Doppler shift frequency and the decay
constant are dependent on the geometry
of the optical system and are expressed
by the equations.
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Electrophoretic light scattering - Heterodyne light scattering
where \upsilon_D \, is velocity of the
particles and \ D \, is the diffusion
constant|translational diffusion constant of
particles.
1
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Electrophoretic light scattering - Heterodyne light scattering
1
The amplitude of the scattering
vector \ q \, is given by the
equation
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Electrophoretic light scattering - Heterodyne light scattering
1
Since velocity \ V \, is proportional
to the applied electric field, \ E \,, the
apparent electrophoretic mobility \
\mu_ \, is define by the equation
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Electrophoretic light scattering - Heterodyne light scattering
Finally, the relation between the
Doppler shift frequency and mobility is
given for the case of the optical
configuration of Fig. 3 by the equation
1
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Electrophoretic light scattering - Heterodyne light scattering
1
:
\upsilon_D = \mu_ \frac \sin
\theta \qquad(7)
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Electrophoretic light scattering - Heterodyne light scattering
1
where \ E \, is the strength of the electric
field, \ n\, the refractive index of the
medium, \ \lambda_0 \,, the wavelength of
the incident light in vacuum, and \ \theta \,
the scattering angle.
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Electrophoretic light scattering - Heterodyne light scattering
1
The sign of \ v_D \, is a result of vector
calculation and depends on the geometry
of the optics.
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Electrophoretic light scattering - Heterodyne light scattering
1
The spectral frequency can be
obtained according to Eq. (2).
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Electrophoretic light scattering - Heterodyne light scattering
1
When \ | \upsilon_M |
gallery/gallery
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Light scattering
1
Deviations from the law of reflection
due to irregularities on a surface are
also usually considered to be a form of
scattering
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Light scattering
1
Most objects that one sees are visible due
to light scattering from their surfaces.
Indeed, this is our primary mechanism of
physical observation.
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Light scattering
Scattering of light depends on the
wavelength or frequency of the light
being scattered. Since visible light has
wavelength on the order of a
micrometre, objects much smaller than
this cannot be seen, even with the aid
of a microscope. Colloidal particles as
small as 1µm have been observed
directly in aqueous suspension.
1
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Light scattering
The transmission
coefficient#Optics|transmission of various
frequencies of light is essential for
applications ranging from window glass to
optical fiber cable|fiber optic transmission
cables and infrared (IR) heat-seeking
missile detection systems. Light
propagating through an optical system can
be attenuation|attenuated by Absorption
(electromagnetic radiation)|absorption,
1
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Light scattering - Introduction
1
With regard to light scattering in liquids and
solids, primary material considerations
include:
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Light scattering - Introduction
*Crystalline structure: How closepacked its atoms or molecules are,
and whether or not the atoms or
molecules exhibit the long-range
order evidenced in crystalline solids.
1
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Light scattering - Introduction
1
*Glassy structure: Scattering centers include
fluctuations in density and/or composition.
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Light scattering - Introduction
1
*Microstructure: Scattering centers
include internal surfaces in liquids
due largely to density fluctuations,
and microstructural defects in solids
such as grains, grain boundaries, and
microscopic pores.
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Light scattering - Introduction
1
In the process of light scattering, the
most critical factor is the length scale
of any or all of these structural features
relative to the wavelength of the light
being scattered.
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Light scattering - Introduction
It has therefore become quite clear that
light scattering is an extremely useful tool
for monitoring the dynamics of structural
relaxation in glasses on various temporal
and spatial scales and therefore provides
an ideal tool for quantifying the capacity of
various glass compositions for guided light
wave transmission well into the far infrared
portions of the electromagnetic spectrum.
1
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Light scattering - Introduction
1
*Note: Light scattering in an ideal defectfree crystalline (non-metallic) solid which
provides no scattering centers for
incoming lightwaves will be due primarily
to any effects of anharmonicity within the
ordered lattice
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Light scattering - Types of scattering
1
Rayleigh scattering is the main
cause of signal loss in optical
fibers.I
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Light scattering - Types of scattering
Mie scattering intensity for large
particles is proportional to the square
of the particle diameter.
1
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Light scattering - Types of scattering
*Tyndall scattering is similar to Mie
scattering without the restriction to
spherical geometry of the particles. It is
particularly applicable to colloidal mixtures
and Suspension (chemistry)|suspensions.
1
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Light scattering - Types of scattering
1
Brillouin scattering measurements
yield the sound velocities in a
material, which may be used to
calculate the elastic constants of the
sample.
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Light scattering - Types of scattering
1
Raman scattering may therefore be used to
determine chemical composition and
molecular structure
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Light scattering - Static and dynamic scattering
A common dichotomy in light scattering
terminology is static light scattering versus
dynamic light scattering. In static light
scattering, the experimental variable is the
time-average intensity of scattered light,
whereas in dynamic light scattering it is
the fluctuations in light intensity that are
studied. Both techniques are typically
encountered in the field of colloid and
polymer characterization. They also have
1
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Light scattering - Critical phenomena
1
Density fluctuations are responsible for the
phenomenon of critical opalescence,
which arises in the region of a continuous,
or second-order, phase transition
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Protein crystallography - Scattering
As described in the #Diffraction
theory|mathematical derivation below, the
X-ray scattering is determined by the
density of electrons within the crystal.
Since the energy of an X-ray is much
greater than that of a valence electron, the
scattering may be modeled as Thomson
scattering, the interaction of an
electromagnetic ray with a free electron.
This model is generally adopted to
1
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Protein crystallography - Scattering
Hence the atomic nuclei, which are
much heavier than an electron,
contribute negligibly to the scattered Xrays.
1
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Electron beam lithography - Scattering
These electrons are called
'backscattering|backscattered electrons'
and have the same effect as long-range
Lens flare|flare in optical projection
systems
1
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Neutron - Neutron detection by elastic scattering
1
Detectors relying on
elastic scattering are
called fast neutron
detectors
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Neutron - Neutron detection by elastic scattering
Fast neutron detectors have the
advantage of not requiring a
moderator, and therefore being
capable of measuring the neutron's
energy, time of arrival, and in certain
cases direction of incidence.
1
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X-ray - Compton scattering
1
The transferred energy can be directly
obtained from the scattering angle from
the conservation of energy and
conservation of
momentum|momentum.
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Neutron cross-section - Scattering cross-section
1
The scattering cross-section can be
further subdivided into coherent
scattering and incoherent scattering,
which is caused by the spin
(physics)|spin dependence of the
scattering cross-section and for a
natural sample, presence of different
isotopes of the same element in the
sample.
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Neutron cross-section - Scattering cross-section
1
Since neutrons interact with the nuclear
potential, the scattering cross-section
varies for different isotopes of the element
in question. A very prominent example is
hydrogen and its isotope deuterium. The
total cross-section for hydrogen is over 10
times that of deuterium, mostly due to the
large incoherent scattering length of
hydrogen. Metals tend to be rather
transparent to neutrons, aluminum and
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Sonar - Scattering
1
For analogous reasons active sonar needs to
transmit in a narrow beam to minimise
scattering.
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Giant magnetoresistance - Spin-dependent scattering
Scattering depends on the relative
orientations of the electron spins and
those magnetic moments: it is weakest
when they are parallel and strongest when
they are antiparallel; it is relatively strong
in the paramagnetic state, in which the
magnetic moments of the atoms have
random orientations.
1
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Giant magnetoresistance - Spin-dependent scattering
The hybridized spd band has a high
density of states, which results in stronger
scattering and thus shorter mean free path
λ for minority-spin than majority-spin
electrons
1
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Giant magnetoresistance - Spin-dependent scattering
According to the Drude theory, the
conductivity is proportional to λ, which
ranges from several to several tens of
nanometers in thin metal films
1
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Giant magnetoresistance - Spin-dependent scattering
In some materials, the interaction
between electrons and atoms is the
weakest when their magnetic moments
are antiparallel rather than parallel. A
combination of both types of materials
can result in a so-called inverse GMR
effect.
1
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Cross section (physics) - Scattering
1
In terms of area, the total crosssection (σ) is the sum of the crosssections due to absorption cross
section|absorption, scattering and
luminescence
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Cross section (physics) - Scattering
The total cross-section is related to the
absorbance of the light intensity through
Beer-Lambert|Beer-Lambert's law, which
says absorbance is proportional to
concentration: A_\lambda = C \,\ell\,
\sigma, where C is the concentration as a
number density, Aλ is the absorbance at a
given wavelength λ, and \ell is the path
length
1
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Optical properties of carbon nanotubes - Raman scattering
Raman spectroscopy has good spatial
resolution (~0.5 micrometers) and
sensitivity (single nanotubes); it requires
only minimal sample preparation and is
rather informative. Consequently, Raman
spectroscopy is probably the most popular
technique of carbon nanotube
characterization. Raman scattering in
SWCNTs is resonant, i.e., only those tubes
are probed which have one of the
1
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Optical properties of carbon nanotubes - Raman scattering
Several scattering
modes dominate the
SWCNT spectrum, as
discussed below.
1
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Optical properties of carbon nanotubes - Raman scattering
1
Similar to photoluminescence mapping,
the energy of the excitation light can be
scanned in Raman measurements, thus
producing Raman maps. Those maps also
contain oval-shaped features uniquely
identifying (n,m) indices. Contrary to PL,
Raman mapping detects not only
semiconducting but also metallic tubes,
and it is less sensitive to nanotube
bundling than PL. However, requirement
of a tunable laser and a dedicated
spectrometer is a strong technical
impediment.
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Optical properties of carbon nanotubes - Anti-Stokes scattering
All the above Raman modes can be
observed both as Raman
scattering#Raman_scattering:_Stokes_an
d_anti-Stokes|Stokes and anti-Stokes
scattering. As mentioned above, Raman
scattering from CNTs is resonant in nature,
i.e. only tubes whose band gap energy is
similar to the laser energy are excited. The
difference between those two energies,
and thus the band gap of individual tubes,
1
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Optical properties of carbon nanotubes - Anti-Stokes scattering
This estimate however relies on the
temperature factor (Boltzmann factor),
which is often miscalculated – focused
laser beam is used in the measurement,
which can locally heat the nanotubes
without changing the overall temperature
of the studied sample.
1
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Optical properties of carbon nanotubes - Rayleigh scattering
Carbon nanotubes have very large
aspect ratio, i.e., their length is much
larger than their diameter.
Consequently, as expected from the
Maxwell's equations|classical
electromagnetic theory, elastic light
scattering (or Rayleigh scattering) by
straight CNTs has anisotropic angular
dependence, and from its spectrum,
the band gaps of individual nanotubes
1
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Optical properties of carbon nanotubes - Rayleigh scattering
Another manifestation of Rayleigh
scattering is the antenna effect, an array of
nanotubes standing on a substrate has
specific angular and spectral distributions
of reflected light, and both those
distributions depend on the nanotube
length.
1
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Ballistic conduction - Scattering mechanisms
In general, carriers will exhibit
ballistic conduction when L \le
\lambda_ where L is the length of the
active part of the device (i.e., a
channel in a MOSFET). \lambda_ is
the mean scattering length for the
carrier which can be given by
Matthiessen's rule#Matthiessen's
rule|Matthiessen's Rule, written here
for electrons:
1
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Ballistic conduction - Scattering mechanisms
1
\frac \lambda_\mathrm
MFPmain|Landauer
formulacite book
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Ballistic conduction - Scattering mechanisms
1
|author=Supriyo Datta, Contributor: Haroon
Ahmad, Alec Broers, Michael Pepper
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Ballistic conduction - Scattering mechanisms
1
|url=http://books.google.com/?id=28
BCofEhvUCdq=electronic+transport+in+
mesoscopic+systemsprintsec=frontco
verCitation needed|date=June 2010
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X-ray scattering techniques
1
Note that X-ray scattering is different from
X-ray diffraction, which is widely used for
X-ray crystallography.
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X-ray scattering techniques - Elastic scattering
1
Materials that do not have long range
order may also be studied by scattering
methods that rely on elastic
collision|elastic scattering of
monochromatic X-rays.
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X-ray scattering techniques - Elastic scattering
* Small-angle X-ray scattering (SAXS)
probes structure in the nanometer to
micrometer range by measuring scattering
intensity at scattering angles 2θ close to
0°.
1
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X-ray scattering techniques - Elastic scattering
* X-ray reflectivity is an analytical
technique for determining thickness,
roughness, and density of single layer
and multilayer thin films.
1
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X-ray scattering techniques - Elastic scattering
1
* Wide-angle X-ray scattering (WAXS),
a technique concentrating on
scattering angles 2θ larger than 5°.
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X-ray scattering techniques - Inelastic scattering
1
When the energy and angle of the inelastic
collision|inelastically scattered X-rays are
monitored, scattering techniques can be
used to probe the electronic band
structure of materials. Inelastic scattering
alters the phase of the diffracted x-rays,
and as a result do not produce useful data
for x-ray diffraction. Rather, inelastically
scattered x-rays contribute to the
background noise in a diffraction pattern.
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Subsurface scattering
1
Subsurface scattering is important in 3D
computer graphics, being necessary for
the realistic rendering of materials such as
marble, skin, leaf|leaves, wax and milk.
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Subsurface scattering - Rendering Techniques
Most materials used in real-time
computer graphics today only account
for the interaction of light at the
surface of an object. In reality, many
materials are slightly translucent:
light enters the surface; is absorbed,
scattered and re-emitted— potentially
at a different point. Skin is a good case
in point; only about 6% of reflectance
is direct, 94% is from subsurface
1
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Subsurface scattering - Rendering Techniques
An inherent property of
semitransparent materials is
absorption. The further through the
material light travels, the greater the
proportion absorbed. To simulate this
effect, a measure of the distance the
light has traveled through the
material must be obtained.
1
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Subsurface scattering - Depth Map based SSS
One method of
estimating this distance
is to use depth maps
1
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Subsurface scattering - Depth Map based SSS
1
, in a manner similar to shadow
mapping
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Subsurface scattering - Depth Map based SSS
1
The measure of distance obtained by
this method can be used in several
ways. One such way is to use it to
index directly into an artist created 1D
texture that falls off exponentially with
distance. This approach, combined
with other more traditional lighting
models, allows the creation of
different materials such as marble,
jade and wax.
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Subsurface scattering - Depth Map based SSS
Potentially, problems can arise if
models are not convex, but depth
peeling can be used to avoid the
issue. Similarly, depth peeling can be
used to account for varying densities
beneath the surface, such as bone or
muscle, to give a more accurate
scattering model.
1
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Subsurface scattering - Depth Map based SSS
1
As can be seen in the image of the wax
head to the right, light isn’t diffused
when passing through object using this
technique; back features are clearly
shown. One solution to this is to take
multiple samples at different points on
surface of the depth map. Alternatively,
a different approach to approximation
can be used, known as texture-space
diffusion.
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Subsurface scattering - Texture Space Diffusion
1
As noted at the start of the section, one of
the more obvious effects of subsurface
scattering is a general blurring of the
diffuse lighting. Rather than arbitrarily
modifying the diffuse function, diffusion
can be more accurately modeled by
simulating it in texture space. This
technique was pioneered in rendering
faces in The Matrix Reloaded, but has
recently fallen into the realm of real-time
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Subsurface scattering - Texture Space Diffusion
1
For human skin, the broadest scattering is in red,
then green, and blue has very little scattering.
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Subsurface scattering - Texture Space Diffusion
A major benefit of this method is its
independence of screen resolution;
shading is performed only once per
texel in the texture map, rather than for
every pixel on the object
1
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Photon mapping - Subsurface scattering
1
Subsurface scattering is the effect
evident when light enters a material
and is scattered before being
absorbed or reflected in a different
direction. Subsurface scattering can
accurately be modeled using photon
mapping. This was the original way
Jensen implemented it; however, the
method becomes slow for highly
scattering materials, and
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Mie theory - Mie scattering codes
There are several known objects which
allow such a solution: spheres, concentric
spheres, infinite cylinders, cluster of
spheres and cluster of cylinders, there are
also known series solutions for scattering
on ellipsoidal particles
1
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Mie theory - Mie scattering codes
* Codes for electromagnetic
scattering by spheres mdash;
solutions for single sphere, coated
spheres, multilayer sphere, cluster of
spheres
1
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Mie theory - Mie scattering codes
1
* Codes for electromagnetic scattering by
cylinders mdash; solutions for single
cylinder, multilayer cylinders, cluster of
cylinders.
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Mie theory - Mie scattering codes
1
A generalization that allows for a treatment
of more general shaped particles is the Tmatrix method, which also relies on the
series approximation to solutions of
Maxwell's equations.
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Mie theory - Rayleigh approximation (scattering)
1
Rayleigh scattering describes the elastic
scattering of light by spheres which are
much smaller than the wavelength of light.
The intensity, I, of the scattered radiation
is given by
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Mie theory - Rayleigh approximation (scattering)
1
It can be seen from the above equation
that Rayleigh scattering is strongly
dependent upon the size of the particle
and the wavelengths. The intensity of the
Rayleigh scattered radiation increases
rapidly as the ratio of particle size to
wavelength increases. Furthermore, the
intensity of Rayleigh scattered radiation is
identical in the forward and reverse
directions.
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Mie theory - Rayleigh approximation (scattering)
It can be shown, however, that Mie
scattering differs from Rayleigh scattering
in several respects; it is roughly
independent of wavelength and it is larger
in the forward direction than in the reverse
direction
1
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Mie theory - Rayleigh approximation (scattering)
During sunrises and sunsets, the
Rayleigh scattering effect is much
more noticeable due to the larger
volume of air through which sunlight
passes.
1
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Mie theory - Rayleigh approximation (scattering)
1
In contrast, the water droplets which
make up clouds are of a comparable
size to the wavelengths in visible
light, and the scattering is described
by Mie's model rather than that of
Rayleigh. Here, all wavelengths of
visible light are scattered
approximately identically and the
clouds therefore appear to be white or
grey.
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Defect scattering
1
The break in periodicity results in a decrease
in conductivity due to 'defect scattering'.
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Defect scattering - Electronic Energy Levels of Semiconductor Dangling Bonds
Compound semiconductors, such as
GaAs, have dangling bond states that are
nearer to the band edges (see Figure 2)
1
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Defect scattering - Electronic Energy Levels of Semiconductor Dangling Bonds
1
where U=-q2/(4πεεrr) is the electrostatic
potential between an electron occupying
the dangling bond and its ion core with ε,
the free space permittivity constant, εr, the
relative permittivity, and r the electron-ion
core separation. The simplification that the
electron translational energy, KE=-U/2, is
due to the virial theorem for
centrosymmetric potentials. As described
by the Bohr model, r is subject to
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Defect scattering - Electronic Energy Levels of Semiconductor Dangling Bonds
1
The electron momentum is
p=mv=h/λ such that
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Condensed matter physics - Scattering
Coulomb and Mott scattering
measurements can be made by using
electron beams as scattering probes,
and similarly, positron annihilation can
be used as an indirect measurement of
local electron density
1
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Optical fibre - Light scattering
1
It has recently been shown that when the
size of the scattering center (or grain
boundary) is reduced below the size of the
wavelength of the light being scattered,
the scattering no longer occurs to any
significant extent
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Optical fibre - Light scattering
1
Distributed both between and within these
domains are micro-structural defects that
provide the most ideal locations for light
scattering
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Mie scattering
1
The 'Mie solution' to Maxwell's equations
(also known as the 'Lorenz–Mie solution',
the 'Lorenz–Mie–Debye solution' or 'Mie
scattering') describes the scattering of
electromagnetic radiation by a sphere.
The solution takes the form of an
analytical infinite series. It is named after
Gustav Mie.
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Mie scattering
Currently, the term Mie solution is also
used in broader contexts, for example
when discussing solutions of Maxwell's
equations for scattering by stratified
spheres or by infinite cylinders, or
generally when dealing with scattering
problems solved using the exact Maxwell
equations in cases where one can write
separation of variables|separate equations
for the radial and angular dependence of
1
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Mie scattering - Introduction
1
In this formulation, the incident plane
wave as well as the scattering field is
expanded into radiating spherical
Vector (geometry)|vector wave
functions
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Mie scattering - Introduction
For particles much larger or much
smaller than the wavelength of the
scattered light there are simple and
excellent approximations that suffice to
describe the behaviour of the system.
But for objects whose size is similar to
the wavelength, e.g., water droplets in
the atmosphere, latex particles in paint,
droplets in emulsions including milk,
and biological cells and cellular
1
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Mie scattering - Introduction
1
The Mie solution [http://diogenes.iwt.unibremen.de/vt/laser/papers/RAE-LT18731976-Mie-1908-translation.pdf English
translation], [http://diogenes.iwt.unibremen.de/vt/laser/papers/SAND78-6018Mie-1908-translation.pdf American
translation] is named after its developer,
German physicist Gustav Mie. Danish
physicist Ludvig Lorenz and others
independently developed the theory of
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Mie scattering - Introduction
1
The existence of resonances and other
features of Mie scattering, make it a
particularly useful formalism when
using scattered light to measure
particle size.
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Mie scattering - Rayleigh Gans Approximation
1
The Rayleigh Gans Approximation is an
approximate solution to light scattering
when the relative refractive index of the
particle is close to unity, and its size is
much smaller in comparison to the
wavelength of light divided by |n−1|, where
n is the refractive index.
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Mie scattering - Anomalous diffraction approximation of van de Hulst
The anomalous diffraction
approximation is valid for large and
optically soft spheres. The extinction
efficiency in this approximation is
given by
1
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Mie scattering - Anomalous diffraction approximation of van de Hulst
1
has as its physical meaning, the phase delay of
the wave passing through the centre of the
sphere;
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Mie scattering - Anomalous diffraction approximation of van de Hulst
1
where a is the sphere radius, n is the ratio
of refractive indices inside and outside of
the sphere, and λ the wavelength of the
light.
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Mie scattering - Anomalous diffraction approximation of van de Hulst
1
This set of equations was first
described by van de Hulst in
(1957).
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Mie scattering - Applications
Mie theory is very important in
meteorology|meteorological optics,
where diameter-to-wavelength ratios of
the order of unity and larger are
characteristic of many problems
regarding haze and cloud scattering. A
further application is in the
characterization of aerosol|particles via
optical scattering measurements. The
Mie solution is also important for
understanding the appearance of
common materials like milk, biological
1
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Mie scattering - Atmospheric science
Mie scattering occurs when the
particles in the atmosphere are the
same size as the wavelengths being
scattered. Dust, pollen, smoke and
microscopic water droplets are
common causes of Mie scattering
which tends to affect longer
wavelengths. Mie scattering occurs
mostly in the lower portions of the
atmosphere where larger particles are
1
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Raman scattering
It was discovered by C. V. Raman and
Kariamanickam Srinivasa Krishnan|K. S. Krishnan in
liquids,
1
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Raman scattering
1
In a gas, Raman scattering can occur with
a change in energy of a molecule due to a
transition (see energy level)
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Raman scattering - History
1
Raman received the Nobel Prize in 1930 for his
work on the scattering of light.
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Raman scattering - History
In 1998 the Raman effect was
designated a National Historic
Chemical Landmarks|National
Historic Chemical Landmark in
recognition of its significance as a tool
for analyzing the composition of
liquids, gases, and solids.
1
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Raman scattering - Degrees of Freedom
1
For any given chemical compound,
there are a total of 3N Degrees of
freedom (physics and
chemistry)|degrees of freedom,
where N is the number of atoms in the
compound
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Raman scattering - Degrees of Freedom
1
whereas for a non-linear
molecule the number of
vibrational modes are
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Raman scattering - Molecular Vibrations and Infrared Radiation
The frequencies of molecular
vibrations range from less than 1012 to
approximately 1014 Hz. These
frequencies correspond to radiation in
the infrared (IR) region of the
electromagnetic spectrum. At any
given instant, each molecule in a
sample has a certain amount of
vibrational energy. However, the
amount of vibrational energy that a
1
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Raman scattering - Molecular Vibrations and Infrared Radiation
At room temperature, most of
molecules will be in the lowest energy
level|energy state, which is known as
the ground state
1
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Raman scattering - Molecular Vibrations and Infrared Radiation
where h is Planck's constant|Planck’s
constant and \nu is the frequency of the
radiation. Thus, the energy required for
such a transition may be calculated if the
frequency of the incident radiation is
known.
1
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Raman Scattering
Other investigations carried out by
Raman were: his experimental and
theoretical studies on the diffraction of
light by acoustic waves of ultrasonic
and hypersonic frequencies (published
1934-1942), and those on the effects
produced by X-rays on infrared
vibrations in crystals exposed to
ordinary light
1
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Raman Scattering
It is also possible to observe
molecular vibrations by an inelastic
scattering process. In inelastic
scattering, an absorbed photon is reemitted with lower energy. In Raman
scattering, the difference in energy
between the excitation and scattered
photons corresponds to the energy
required to excite a molecule to a
higher vibrational mode.
1
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Raman Scattering
1
This type of scattering is known as
Rayleigh scattering.
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Raman Scattering
1
However, when the polarization in the
molecules couples to a vibrational state
that is higher in energy than the state
they started in, then the original photon
and the scattered photon differ in
energy by the amount required to
vibrationally excite the molecule
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Raman scattering - Stokes and anti-Stokes scattering
1
The Raman interaction leads to
two possible outcomes:
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Raman scattering - Stokes and anti-Stokes scattering
1
*the material absorbs energy and the
emitted photon has a lower energy than
the absorbed photon. This outcome is
labeled Stokes line|Stokes Raman
scattering.
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Raman scattering - Stokes and anti-Stokes scattering
1
*the material loses energy and the emitted
photon has a higher energy than the
absorbed photon. This outcome is labeled
anti-Stokes Raman scattering.
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Raman scattering - Stokes and anti-Stokes scattering
1
The energy difference between the
absorbed and emitted photon
corresponds to the energy difference
between two resonant states of the
material and is independent of the
absolute energy of the photon.
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Raman scattering - Stokes and anti-Stokes scattering
1
Correspondingly, Stokes scattering peaks are
stronger than anti-Stokes scattering peaks
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Raman scattering - Distinction from fluorescence
1
The Raman effect
differs from the
process of
fluorescence
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Raman scattering - Selection rules
1
The distortion of a molecule in an electric
field, and therefore the vibrational Raman
cross section (physics)|cross section, is
determined by its polarizability
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Raman scattering - Stimulated Raman scattering and Raman amplification
The Raman-scattering process as
described above takes place
spontaneously; i.e., in random time
intervals, one of the many incoming
photons is scattered by the material.
This process is thus called
spontaneous Raman scattering.
1
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Raman scattering - Stimulated Raman scattering and Raman amplification
In that case, the total Ramanscattering rate is increased beyond
that of spontaneous Raman scattering:
pump photons are converted more
rapidly into additional Stokes photons
1
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Raman scattering - Stimulated Raman scattering and Raman amplification
Stimulated Raman scattering is a
Nonlinear optics|nonlinear-optical effect. It
can be described, e.g., using a third-order
nonlinear susceptibility \chi^.
1
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Raman scattering - Need of coherence
1
Suppose that the distance between two
points A and B of an exciting beam is .
Generally, as the exciting frequency is
not equal to the scattered Raman
frequency, the corresponding relative
wavelengths and are not equal. Thus,
a phase-shift appears. For , the
scattered amplitudes are opposite, so
that the Raman scattered beam
remains weak.
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Raman scattering - Need of coherence
1
Several tricks may be used
to get a large amplitude:
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Raman scattering - Need of coherence
1
- Conveniently used optically anisotropic crystals
may have an equal relative wavelength
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Raman scattering - Need of coherence
1
- Light may be pulsed, so
that beats do not appear.
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Raman scattering - Need of coherence
1
It is the Impulsive Stimulated Raman Scattering
(ISRS), in which the length of the pulses
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Raman scattering - Need of coherence
must be shorter than all relevant time
constants. The interference of the Raman
and incident light is too short to allow
beats, so that it produces a frequency
shift. In labs, femtosecond laser pulses
must be used because the ISRS becomes
very weak if the pulses are too long. Thus
ISRS cannot be observed using ordinary
incoherent light. However, it appears in
space, in excited atomic hydrogen,
producing Hubble's redshift.
1
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Raman scattering - Applications
Raman spectroscopy employs the
Raman effect for materials analysis.
The spectrum of the Raman-scattered
light depends on the molecular
constituents present and their state,
allowing the spectrum to be used for
material identification and analysis.
Raman spectroscopy is used to
analyze a wide range of materials,
including gases, liquids, and solids.
1
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Raman scattering - Applications
1
For solid materials, Raman scattering
is used as a tool to detect highfrequency phonon and magnon
excitations.
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Raman scattering - Applications
Raman lidar is used in atmospheric
physics to measure the atmospheric
extinction coefficient and the water vapour
vertical distribution.
1
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Raman scattering - Applications
1
Stimulated Raman transitions are also
widely used for manipulating a trapped
ion's energy levels, and thus basis
qubit states.
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Raman scattering - Applications
Raman spectroscopy can be used to
determine the force constant and bond
length for molecules that do not have an
infrared absorption spectrum.
1
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Raman scattering - Applications
1
Raman amplification is used in
optical amplifiers.
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Raman scattering - Supercontinuum generation
1
For high-intensity CW (continuous wave)
lasers, SRS can be used to produce broad
bandwidth spectra
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Polarized - Optical scattering
1
Mueller matrices are then used to describe
the observed polarization effects of the
scattering of waves from complex surfaces
or ensembles of particles, as shall now be
presented.
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Thomson Scattering
'Thomson scattering' is the elastic
scattering of electromagnetic
radiation by a free charged particle,
as described by classical
electromagnetism. It is just the lowenergy limit of Compton scattering:
the particle kinetic energy and photon
frequency are the same before and
after the scattering. This limit is valid
as long as the photon energy is much
less than the mass energy of the
particle: \nu\ll mc^2/h .
1
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Thomson Scattering - Introduction
1
Thomson scattering is an important
phenomenon in plasma physics and
was first explained by the physicist J.J
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Thomson Scattering - Introduction
The electric fields of the incoming and
observed beam can be divided up into
those components lying in the plane of
observation (formed by the incoming and
observed beams) and those components
perpendicular to that plane. Those
components lying in the plane are referred
to as radial and those perpendicular to the
plane are tangential, since this is how they
appear to the observer.
1
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Thomson Scattering - Introduction
It shows the radial component of the
incident electric field causing a component
of motion of the charged particles at the
scattering point which also lies in the plane
of observation
1
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Thomson Scattering - Introduction
The scattering is best described by an
emission coefficient which is defined as ε
where ε dt dV dΩ dλ is the energy
scattered by a volume element dV in time
dt into solid angle dΩ between
wavelengths λ and λ+dλ. From the point of
view of an observer, there are two
emission coefficients, εr corresponding to
radially polarized light and εt
corresponding to tangentially polarized
1
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Thomson Scattering - Introduction
1
where n is the density of charged particles
at the scattering point, I is incident flux (i.e.
energy/time/area/wavelength) and
\sigma_t is the Thomson Cross section
(physics)|cross section for the charged
particle, defined below. The total energy
radiated by a volume element dV in time
dt between wavelengths λ and λ+dλ is
found by integrating the sum of the
emission coefficients over all directions
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Thomson Scattering - Introduction
1
The Thomson differential cross section,
related to the sum of the emissivity
coefficients, is given by
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Thomson Scattering - Introduction
1
where the first expression is in Centimeter
gram second system of units|cgs units, the
second in SI units; q is the charge per
particle, m the mass of particle, and
\epsilon_0 a constant, the permittivity of
free space. Integrating over the solid
angle, we obtain the Thomson cross
section (in cgs and SI units):
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Thomson Scattering - Introduction
The important feature is that the cross
section is independent of photon
frequency. Note that the cross section is
simply proportional (by a numerical factor)
to the square of the classical electron
radius|classical radius of a point particle of
mass m and charge q:
1
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Thomson Scattering - Introduction
1
Alternatively, this can be seen in terms of
\lambda_c, the Compton wavelength, and
the Coupling constant|fine structure
constant:
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Thomson Scattering - Introduction
For an electron, the
Thomson crosssection is numerically
given by:
1
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Thomson Scattering - Examples of Thomson scattering
1
The cosmic microwave background is
linearly polarized as a result of
Thomson scattering, as measured by
Degree Angular Scale
Interferometer|DASI and more recent
experiments.
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Thomson Scattering - Examples of Thomson scattering
1
The solar K-corona is the result of the
Thomson scattering of solar radiation
from solar coronal electrons. NASA's
STEREO mission generates threedimensional images of the electron
density around the sun by measuring
this K-corona from two separate
satellites.
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Thomson Scattering - Examples of Thomson scattering
In tokamaks and other experimental
fusion power|fusion devices, the
electron temperatures and densities in
the plasma (physics)|plasma can be
plasma diagnostics#Thomson
scattering|measured with high
accuracy by detecting the effect of
Thomson scattering of a high-intensity
laser beam.
1
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Molecular weight - Static light scattering
The only external measurement
required is Static light
scattering#Theory|refractive index
increment, which describes the change
in refractive index with concentration.
1
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Transparency (optics) - Light scattering in solids
Light scattering in liquids and
solids|Light scattering from the
surfaces of objects is our primary
mechanism of physical observation.
1
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Transparency (optics) - Light scattering in solids
1
Scattering centers (or particles) as
small as one micrometer have been
observed directly in the light
microscope (e.g., Brownian motion).
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Neutron diffraction - Nuclear scattering
Like all quantum elementary
particle|particles, neutrons can
exhibit wave phenomena typically
associated with light or sound
1
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Neutron diffraction - Nuclear scattering
1
The scattering length varies from isotope to
isotope rather than linearly with the atomic
number
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Neutron diffraction - Nuclear scattering
1
Furthermore, there is no need for an
atomic form factor to describe the
shape of the electron cloud of the atom
and the scattering power of an atom
does not fall off with the scattering
angle as it does for X-rays
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Neutron diffraction - Magnetic scattering
Although neutrons are uncharged,
they carry a spin, and therefore
interact with magnetic moments,
including those arising from the
electron cloud around an atom.
Neutron diffraction can therefore
reveal the microscopic magnetic
structure of a material.Neutron
diffraction of magnetic materials / Yu.
A. Izyumov, V.E. Naish, and R.P.
1
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Neutron diffraction - Magnetic scattering
1
Magnetic scattering does require an
atomic form factor#Magnetic
scattering|atomic form factor as it is
caused by the much larger electron
cloud around the tiny nucleus. The
intensity of the magnetic contribution
to the diffraction peaks will therefore
dwindle towards higher angles.
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Neutron diffraction - Hydrogen, null-scattering and contrast variation
Neutron diffraction can be used to
establish the structure of low atomic
number materials like proteins and
surfactants much more easily with
lower flux than at a synchrotron
radiation source. This is because some
low atomic number materials have a
higher cross section for neutron
interaction than higher atomic weight
materials.
1
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Neutron diffraction - Hydrogen, null-scattering and contrast variation
1
The greater scattering power of protons
and deuterons means that the position of
hydrogen in a crystal and its thermal
motions can be determined with greater
precision by neutron diffraction
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Neutron diffraction - Hydrogen, null-scattering and contrast variation
The neutron scattering lengths bH = 3.7406(11) fm and bD = 6.671(4) fm, for H
and D respectively, have opposite sign,
which allows the technique to distinguish
them. In fact there is a particular isotope
ratio for which the contribution of the
element would cancel, this is called nullscattering.
1
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Neutron diffraction - Hydrogen, null-scattering and contrast variation
1
Nevertheless, by preparing samples with
different isotope ratios it is possible to vary
the scattering contrast enough to highlight
one element in an otherwise complicated
structure
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Bragg's law - Bragg scattering of visible light by colloids
1
A colloidal crystal is a highly Order (crystal
lattice)|ordered array of particles which
can be formed over a very long range
(from a few millimeters to one centimeter)
in length, and which appear analogous to
their atomic or molecular counterparts
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Bragg's law - Bragg scattering of visible light by colloids
1
In all of these cases in nature, the same
brilliant iridescence (or play of colours) can
be attributed to the diffraction and
constructive interference of visible
lightwaves which satisfy Bragg’s law, in a
matter analogous to the scattering of Xrays in crystalline solid.
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Scattering
1
Reflections that undergo scattering are
often called diffuse reflections and
unscattered reflections are called
specular (mirror-like) reflections
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Scattering
'Scattering' may also refer to particleparticle collisions between molecules,
atoms, electrons, photons and other
particles. Examples are: cosmic rays
scattering by the Earth's upper
atmosphere; particle collisions inside
particle accelerators; electron scattering
by gas atoms in fluorescent lamps; and
neutron scattering inside nuclear reactors.
1
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Scattering
1
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.
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Scattering
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 computergenerated imagery. Particle-particle
scattering theory is important in areas
such as particle physics, atomic,
molecular, and optical physics, nuclear
physics and astrophysics.
1
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Scattering - Single and multiple scattering
Single scattering is
therefore often
described by
probability distributions.
1
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Scattering - Single and multiple scattering
Coherent backscattering, an
enhancement of backscattering that
occurs when coherent radiation is
multiply scattered by a random
medium, is usually attributed to weak
localization.
1
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Scattering - Single and multiple scattering
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 radar scattering as well,
where the targets tend to be macroscopic
objects such as people or aircraft.
1
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Scattering - Single and multiple scattering
In certain rare circumstances,
multiple scattering may only involve a
small number of interactions such
that the randomness is not completely
averaged out
1
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Scattering - Single and multiple scattering
1
The description of scattering and the
distinction between single and multiple
scattering are often highly involved with
wave–particle duality.
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Scattering - Scattering theory
1
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.
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Scattering - Electromagnetic scattering
Inelastic scattering includes Brillouin
scattering, Raman scattering, inelastic Xray scattering and Compton scattering.
1
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Scattering - Electromagnetic scattering
1
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
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Scattering - Electromagnetic scattering
However, resonant light scattering in
nanoparticles can produce many different
highly saturated and vibrant hues,
especially when surface plasmon
resonance is involved (Roqué et al
1
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Scattering - Electromagnetic scattering
where π Dp is the circumference of a
particle and λ is the wavelength of incident
radiation. Based on the value of α, these
domains are:
1
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Scattering - Electromagnetic scattering
1
\alpha \approx 1: Mie scattering
(particle about the same size as
wavelength of light, valid only for
spheres)
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Scattering - Electromagnetic scattering
1
\alpha \gg 1: Geometric scattering (particle much
larger than wavelength of light)
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Scattering - Electromagnetic scattering
1
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
(waves)|polarization, angle, and
Coherence (physics)|coherence.
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Scattering - Electromagnetic scattering
1
Closed-form solutions for scattering by
certain other simple shapes exist, but no
general closed-form solution is known for
arbitrary shapes.
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Scattering - Electromagnetic scattering
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
1
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Scattering - Electromagnetic scattering
1
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.
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Scattering - Electromagnetic scattering
1
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
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Powder diffraction - Comparison of X-ray and neutron scattering
1
In contrast, the neutron scattering lengths of most
atoms are approximately equal in magnitude
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Powder diffraction - Comparison of X-ray and neutron scattering
The incoherent scattering length of
deuterium is much smaller (2.05(3) barn)
making structural investigations
significantly easier
1
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Powder diffraction - Comparison of X-ray and neutron scattering
As neutrons also have a magnetic
moment, they are additionally scattered
by any magnetic moments in a sample.
In the case of long range magnetic
order, this leads to the appearance of
new Bragg reflections. In most simple
cases, powder diffraction may be used
to determine the size of the moments
and their spatial orientation.
1
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Contrast media - Ultrasound scattering and frequency shift
1
This process of backscattering gives
the liquid with these bubbles a high
signal, which can be seen in the
resulting image.
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Small-angle scattering
'Small-angle scattering' (SAS) is a
scattering technique based on
deflection of collimated radiation
away from the straight trajectory after
it interacts with structures that are
much larger than the wavelength of
the radiation. The deflection is small
(0.1-10°) hence the name small-angle.
SAS techniques can give information
about the size, shape and orientation
1
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Small-angle scattering
Small-angle scattering (SAS) is a
powerful technique for investigating
large-scale structures from 10 Å up to
thousands and even several tens of
thousands of angstroms. The most
important feature of the SAS method is
its potential for analyzing the inner
structure of disordered systems, and
frequently the application of this method
is a unique way to obtain direct
structural information on systems with
random arrangement of density
inhomogeneities in such large-scales.
1
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Small-angle scattering
1
Currently, the SAS technique, with its
well-developed experimental and
theoretical procedures and wide range
of studied objects, is a self-contained
branch of the structural analysis of
matter. Reflecting these situations, the
international meeting on the smallangle scattering studies have been held
in every three years.
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Small-angle scattering
1
* Small angle neutron
scattering (SANS)
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Small-angle scattering - Applications
1
Small-angle scattering is particularly
useful because of the dramatic
increase in forward scattering that
occurs at phase transitions, known as
critical opalescence, and because
many materials, substances and
biological systems possess
interesting and complex features in
their structure, which match the
useful length scale ranges that these
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Small-angle scattering - Continuum description
1
small particles in liquid suspension, the
only contrast leading to scattering in the
typical range of resolution of the SAXS is
simply Δρ, the difference in average
electron density between the particle and
the surrounding liquid, because variations
in ρ due to the atomic structure only
become visible at higher angles in the
WAXS regime
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Small-angle scattering - Continuum description
1
SANS is described in
terms of a neutron
scattering length
density.
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Small-angle scattering - Porod's law
1
At wave numbers that are relatively
large on the scale of SAS, but still
small when compared to wide-angle
Bragg diffraction, local interface
intercorrelations are probed, whereas
correlations between opposite
interface segments are averaged out.
For smooth interfaces, one obtains
Porod's law:
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Small-angle scattering - Porod's law
1
This allows the surface area S of the
particles to be determined with SAS.
This needs to be modified if the
interface is rough on the scale q−1. If
the Surface roughness|roughness can
be described by a fractal dimension d
between 2-3 then Porod's law
becomes:
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Small-angle scattering - Scattering from particles
1
Small-angle scattering from particles can
be used to determine the particle shape or
their size distribution. A small-angle
scattering pattern can be fitted with
intensities calculated from different model
shapes when the size distribution is
known. If the shape is known, a size
distribution may be fitted to the intensity.
Typically one assumes the particles to be
sphere|spherical in the latter case.
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Small-angle scattering - Scattering from particles
1
One can write for the
small-angle X-ray
scattering intensity:
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Small-angle scattering - Scattering from particles
1
*I(q) is the intensity as a function of the
magnitude q of the scattering vector
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Small-angle scattering - Scattering from particles
1
*and S(q) is the
structure factor.
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Small-angle scattering - Scattering from particles
One can then easily apply the 'Guinier
approximation' (also called Guinier law,
André Guinier 1911-2000), which applies
only at the very beginning of the scattering
curve, at small q-values
1
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Small-angle scattering - Scattering from particles
1
An important part of the particle shape
determination is usually the 'distance
distribution function' p(r), which may
be calculated from the intensity using a
Fourier transform
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Small-angle scattering - Scattering from particles
1
The distance distribution function p(r) is
related to the frequency of certain
distances r within the particle
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Small-angle scattering - Scattering from particles
1
The particle shape analysis is especially
popular in biological small-angle X-ray
scattering, where one determines the
shapes of proteins and other natural
colloidal polymers.
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Small-angle scattering - History
Small-angle scattering
studies were initiated by
André Guinier (1937).A
1
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Small-angle scattering - Organisations
1
As a 'low resolution' diffraction technique,
the worldwide interests of the small-angle
scattering community are promoted and
coordinated by the
[http://www.iucr.org/resources/commission
s/small-angle-scattering Commission on
Small-Angle Scattering] of the
[http://www.iucr.org/ International Union of
Crystallography]
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Nuclear reaction - Inelastic scattering
1
*(α,α') measures nuclear surface shapes
and sizes. Since α particles that hit the
nucleus react more violently, elastic
scattering|elastic and shallow inelastic α
scattering are sensitive to the shapes and
sizes of the targets, like light
scattering|light scattered from a small
black object.
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Nuclear reaction - Inelastic scattering
*(e,e') is useful for probing the
interior structure. Since electrons
interact less strongly than do protons
and neutrons, they reach to the
centers of the targets and their wave
functions are less distorted by passing
through the nucleus.
1
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Pale Blue Dot - Effects of polarization and scattering of light
1
The degree of polarization for Earth to
appear as a pale blue dot at 443
nanometers and 90° scattering angle,
as calculated using polarization
observations of Earth made from the
POLDER satellite radiometer, is 23%
for 55% (average) cloud cover and up
to 40% for 10% (minimal) cloud cover.
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Radio propagation - Meteor scattering
1
Meteor scattering relies on reflecting
radio waves off the intensely ionized
columns of air generated by meteors
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Radio propagation - Tropospheric scattering
At VHF and higher frequencies, small
variations (turbulence) in the density of
the troposphere|atmosphere at a height of
around 6 miles (10km) can scatter some of
the normally line-of-sight beam of radio
frequency energy back toward the
ground, allowing over-the-horizon
communication between stations as far as
500 miles (800km) apart. The military
developed the White Alice
Communications System covering all of
Alaska, using this tropospheric scattering
principle.
1
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Radio propagation - Rain scattering
Scattering from snowflakes and ice
pellets also occurs, but scattering
from ice without watery surface is less
effective
1
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Radio propagation - Airplane scattering
Airplane scattering (or most often
reflection) is observed on VHF through
microwaves and, besides back-scattering,
yields momentary propagation up to
500km even in mountainous terrain. The
most common back-scatter applications
are air-traffic radar, bistatic forward-scatter
guided-missile and airplane-detecting tripwire radar, and the US space radar.
1
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Radio propagation - Lightning scattering
1
Lightning scattering has sometimes been observed
on VHF and UHF over distance of about 500km
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Absorption spectrum - Relation to scattering and reflection spectra
1
Therefore, the absorption spectrum can be derived
from a scattering or reflection spectrum
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Planetary migration - Gravitational scattering
1
Another possible mechanism that may
move planets over large orbital radii
is gravitational scattering by larger
planets or, in a protoplantetary disk,
gravitational scattering by overdensities in the fluid of the disk
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Open-pool Australian lightwater reactor - Neutron scattering at OPAL
The Bragg Institute at ANSTO hosts
OPAL's neutron scattering facility. It is
now running as a user facility serving
the scientific community in Australia
and around the world. New funding was
received in 2009 in order to install
further competitive instruments and
beamlines. The actual facility
comprises the following instruments:
1
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X-ray diffraction - Elastic vs. inelastic scattering
1
Inelastic scattering is useful for probing
such excitations of matter, but not in
determining the distribution of scatterers
within the matter, which is the goal of Xray crystallography.
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X-ray diffraction - Elastic vs. inelastic scattering
1
Therefore, X-rays are the sweetspot for
wavelength when determining atomicresolution structures from the
scattering of electromagnetic radiation.
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Kramers–Kronig relation - Hadronic Scattering
They are also used under the name
integral dispersion relations with reference
to hadronic scattering. In this case, the
function is the scattering amplitude and
through the use of the optical theorem the
imaginary part of the scattering amplitude
is related to the total Cross section
(physics)|cross section which is a
physically measurable quantity.
1
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Compton scattering
'Compton scattering' is an inelastic
scattering of a photon by a quasi-free
electric charge|charged particle,
usually an electron. It results in a
decrease in energy (increase in
wavelength) of the photon (which may be
an X-ray or gamma ray photon), called
the 'Compton effect'. Part of the energy
of the photon is transferred to the
recoiling electron. 'Inverse Compton
scattering' also exists, in which a
charged particle transfers part of its
1
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Compton scattering - Introduction
1
Compton scattering is an example of
inelastic scattering, because the
wavelength of the scattered light is
different from the incident radiation.
Still, the origin of the effect can be
considered as an elastic collision
between a photon and an electron (In
Compton's original experiment the
energy of the X ray photon (\approx 20
keV) was very much larger than the
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Compton scattering - Introduction
1
The amount the wavelength changes by
is called the 'Compton shift'. Although
nuclear Compton scattering exists,
Compton scattering usually refers to
the interaction involving only the
electrons of an atom. The Compton
effect was observed by Arthur Holly
Compton in 1923 at Washington
University in St. Louis and further
verified by his graduate student Wu
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Compton scattering - Introduction
1
(Classically, light of sufficient intensity for
the electric field to accelerate a charged
particle to a relativistic speed will cause
radiation-pressure recoil and an
associated Doppler shift of the scattered
light,http://www.lle.rochester.edu/media/pu
blications/documents/theses/Moore.pdf
but the effect would become arbitrarily
small at sufficiently low light intensities
regardless of wavelength.) Light must
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Compton scattering - Introduction
1
Experimental verification of momentum
conservation in individual Compton
scattering processes by Bothe and Geiger
as well as by Compton and Simon has
been important in disproving the BKS
theory.
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Compton scattering - Introduction
1
If the photon is of lower energy, but still has
sufficient energy (in general a few eV to a few
keV, corresponding to visible light through
soft X-rays), it can eject an electron from its
host atom entirely (a process known as the
photoelectric effect), instead of undergoing
Compton scattering. Higher energy photons (
and above) may be able to bombard the
nucleus and cause an electron and a positron
to be formed, a process called pair
production.
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Compton scattering - Description of the phenomenon
1
By the early 20th century, research into the
interaction of X-rays with matter was well
under way
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Compton scattering - Description of the phenomenon
1
In his paper, Compton derived the
mathematical relationship between
the shift in wavelength and the
scattering angle of the X-rays by
assuming that each scattered X-ray
photon interacted with only one
electron
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Compton scattering - Description of the phenomenon
The quantity is known as the
Compton wavelength of the electron; it
is equal to . The wavelength shift is at
least zero (for ) and at most twice the
Compton wavelength of the electron
(for ).
1
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Compton scattering - Description of the phenomenon
Compton found that some X-rays
experienced no wavelength shift despite
being scattered through large angles; in
each of these cases the photon failed to
eject an electron. Thus the magnitude of
the shift is related not to the Compton
wavelength of the electron, but to the
Compton wavelength of the entire atom,
which can be upwards of 10thinsp;000
times smaller.
1
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D-brane - D-brane scattering
1
This induces open string production and as a
result the two scattering branes will be
trapped.
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Tyndall effect - Difference from Rayleigh scattering
But, if the colloid particles are
spheroid, Tyndall scattering is
mathematically analysable in terms
of Mie theory, which admits particle
sizes in the rough vicinity of the
wavelength of light.
1
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Tyndall effect - Some phenomena that are not Tyndall scattering
1
On occasion, the term Tyndall effect is
incorrectly applied to light scattering by
macroscopic dust particles in the air.
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Rutherford scattering
1
The classical Rutherford scattering of
alpha particles against gold nuclei is
an example of elastic scattering
because the energy and velocity of the
outgoing scattered particle is the
same as that with which it began.
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Rutherford scattering
1
A similar process probed the insides
of nuclei in the 1960s, and is called
deep inelastic scattering.
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Rutherford scattering
1
The initial discovery was made by Hans
Geiger and Ernest Marsden in 1909 when
they performed the Geiger–Marsden
experiment|gold foil experiment under the
direction of Rutherford, in which they fired
a beam of alpha particles (helium nuclei)
at layers of gold leaf only a few atoms
thick
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Rutherford scattering
However, the intriguing results
showed that around 1 in 8000 alpha
particles were deflected by very large
angles (over 90°), while the rest
passed straight through with little or
no deflection
1
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Rutherford scattering - Derivation
For the case of light alpha particles
scattering off heavy nuclei, as in the
experiment performed by Rutherford,
the reduced mass is essentially the
mass of the alpha particle and the
nucleus off of which it scatters is
essentially stationary in the lab
frame.
1
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Rutherford scattering - Derivation
1
Substituting into the
Binet equation yields
the equation of
trajectory
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Rutherford scattering - : \fracud\theta^ 2anchor|Extension to situations with relativistic
particles and target recoil Extension to situations with relativistic particles and target
recoil
1
The extension of Rutherford-type scattering
to energy regions in which the incoming
particle has spin and magnetic moment, and
is traveling at relativistic energies, and there
is enough momentum-transfer that the struck
particle recoils with some of the incoming
particle's energy (so the process is inelastic
collision|inelastic rather than elastic
collision|elastic), is called Mott
scattering.[http://hyperphysics.phyastr.gsu.edu/hbase/nuclear/elescat.html
Hyperphysics link]
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
RHEED users construct Ewald's spheres to find the
crystallographic properties of the sample surface
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
The Ewald's sphere analysis is similar to
that for bulk crystals, however the
reciprocal lattice for the sample differs
from that for a 3D material due to the
surface sensitivity of the RHEED process
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
The Ewald's sphere is centered on the
sample surface with a radius equal to
the reciprocal of the wavelength of the
incident electrons. The relationship is
given by
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
Diffraction conditions are satisfied
where the rods of reciprocal lattice
intersect the Ewald's sphere.
Therefore, the magnitude of a vector
from the origin of the Ewald's sphere
to the intersection of any reciprocal
lattice rods is equal in magnitude to
that of the incident beam. Equation 2
shows this relationship.
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Reflection high-energy electron diffraction - Kinematic scattering analysis
An arbitrary vector, G, defines the
reciprocal lattice vector between the
ends of any two k vectors. Vector G is
useful for finding distance between
arbitrary planes in the crystal. Vector
G is calculated using Equation 3.
1
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
Figure 3 shows the construction of the
Ewald's sphere and provides
examples of the G, k and k0 vectors.
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Reflection high-energy electron diffraction - Kinematic scattering analysis
Many of the reciprocal lattice rods
meet the diffraction condition,
however the RHEED system is
designed such that only the low orders
of diffraction are incident on the
detector
1
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
The k vectors are labeled such that the k
vector that forms the smallest angle with
the sample surface is called 0th order
beam. The 0th order beam is also known
as the specular beam. Each successive
intersection of a rod and the sphere further
from the sample surface is labeled as a
higher order reflection.
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Reflection high-energy electron diffraction - Kinematic scattering analysis
The center of the Ewald's sphere is
positioned such that the specular beam
forms the same angle with the substrate
as the incident electron beam. The
specular point has the greatest intensity
on a RHEED pattern and is labeled as the
(00) point by convention. The other points
on the RHEED pattern are indexed
according to the reflection order they
project.
1
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Reflection high-energy electron diffraction - Kinematic scattering analysis
The radius of the Ewald's sphere is
much larger than the spacing between
reciprocal lattice rods because the
incident beam has a very short
wavelength due to its high-energy
electrons
1
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
The intersections of these effective planes
with the Ewald's sphere forms circles,
called Laue circles. The RHEED pattern is
a collection of points on the perimeters of
concentric Laue circles around the center
point. However, interference effects
between the diffracted electrons still yield
strong intensities at single points on each
Laue circle. Figure 4 shows the
intersection of one of these planes with the
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
The azimuthal angle affects the
geometry and intensity of
RHEED patterns
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Reflection high-energy electron diffraction - Kinematic scattering analysis
1
Users sometimes rotate the sample
around an axis perpendicular to the
sampling surface during RHEED
experiments to create a RHEED
pattern called the azimuthal plot
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Reflection high-energy electron diffraction - Dynamic scattering analysis
1
The brightness or intensity at a point on
the detector depends on dynamic
scattering, so all analysis involving the
intensity must account for dynamic
scattering
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Neutron scattering
1
Neutron diffraction (elastic scattering) is
used for determining structures; Inelastic
neutron scattering is used for the study of
atomic phonon|vibrations and other
excited state|excitations.
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Neutron scattering - Scattering of fast neutrons
1
At each collision the fast neutron transfers
a significant part of its kinetic energy to the
scattering nucleus; the more so the lighter
the nucleus
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Neutron scattering - Scattering of fast neutrons
1
Neutron moderators are used to produce
thermal neutrons that have kinetic
energies below 1eV (T 500K). Thermal
neutrons are used to maintain a nuclear
chain reaction in a nuclear reactor, and as
a research tool in neutron science
comprising scattering experiments and
other applications (see below). In the
remainder of this article we will
concentrate on the scattering of thermal
neutrons.
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Neutron scattering - Neutron-matter interaction
Since neutrons are electrically neutral,
they penetrate matter more deeply than
electrically charged particles of
comparable kinetic energy; therefore they
are valuable probes of bulk properties.
1
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Neutron scattering - Neutron-matter interaction
Neutron scattering and absorption
Neutron cross-section|cross sections
vary widely from isotope to isotope.
1
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Neutron scattering - Neutron-matter interaction
The nucleus provides a very short
range, isotropic potential varying
randomly from isotope to isotope,
making it possible to tune the nuclear
scattering contrast to suit the
experiment.
1
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Neutron scattering - Neutron-matter interaction
1
The scattering almost always has an
elastic and an inelastic component.
The fraction of elastic scattering is
given by the Debye-Waller factor or
the Mössbauer-Lamb factor.
Depending on the research question,
most measurements concentrate on
either the elastic or the inelastic
scattering.
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Neutron scattering - Neutron-matter interaction
1
Achieving a precise
velocity, i.e
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Neutron scattering - Magnetic scattering
1
The neutron has a net charge of zero,
however its constituent quarks each
have an independent charge. The
triangular arrangement of the three
quarks in a neutron give it a slight
magnetic dipole (similar to the same
dipole phenomenon in water
molecules). The dipole, however, is
quite weak. This also led to the
challenge to initially discover the
neutron as well as the difficulty in
detecting them.
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Neutron scattering - History
The first neutron-scattering
instruments were installed in beam
tubes at multi-purpose research
reactors
1
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Neutron scattering - Facilities
Today, most neutron scattering
experiments are performed by research
scientists who apply for beamtime at
neutron sources through a formal proposal
procedure. Because of the low count rates
involved in neutron scattering
experiments, relatively long periods of
beam time (on the order of days) are
usually required to get good data.
Proposals are assessed for feasibility and
scientific interest.
1
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Neutron scattering - Neutron scattering techniques
1
*Neutron diffraction
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Neutron scattering - Neutron scattering techniques
**Small angle
neutron scattering
1
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Neutron scattering - Neutron scattering techniques
1
*Inelastic neutron
scattering
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Neutron scattering - Neutron scattering techniques
1
**Neutron triple-axis
spectrometry
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Neutron scattering - Neutron scattering techniques
1
**Neutron time-offlight scattering
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Neutron scattering - Neutron scattering techniques
1
**Neutron backscattering
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Enzyme assay - Light Scattering
Static light scattering measures the
product of weight-averaged molar
mass and concentration of
macromolecules in solution. Given a
fixed total concentration of one or
more species over the measurement
time, the scattering signal is a direct
measure of the weight-averaged molar
mass of the solution, which will vary
as complexes form or dissociate.
1
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Enzyme assay - Light Scattering
Hence the measurement quantifies
the stoichiometry of the complexes as
well as kinetics. Light scattering
assays of protein kinetics is a very
general technique that does not
require an enzyme.
1
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Light scattering by particles
1
'Light scattering by particles' is the
process by which small particles such
as ice crystals, dust, planetary dust,
and blood cells cause observable
phenomena such as rainbows, the
color of the sky, and halos.
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Light scattering by particles
1
Maxwell's equations are the basis of
theoretical and computational
methods describing light scattering
but since exact solutions to Maxwell's
equations are only known for selected
geometries (such as spherical
particle) light scattering by particles
is a branch of computational
electromagnetics dealing with
electromagnetic radiation scattering
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Light scattering by particles
Multiple-scattering effects of light
scattering by particles are treated by
radiative transfer techniques (see, e.g
1
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Light scattering by particles - Finite-difference time-domain method
The FDTD method belongs in the
general class of grid-based
differential time-domain numerical
modeling methods
1
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Light scattering by particles - T-matrix
1
The technique is also known as null
field method and extended boundary
technique method (EBCM). Matrix
elements are obtained by matching
boundary conditions for solutions of
Maxwell equations. The incident,
transmitted, and scattered field are
expanded into spherical vector wave
functions.
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Light scattering by particles - Mie approximation
1
Scattering from any spherical particles
with arbitrary size parameter is
explained by the Mie theory. Mie theory,
also called Lorenz-Mie theory or
Lorenz-Mie-Debye theory, is a complete
analytical solution of Maxwell's
equations for the scattering of
electromagnetic radiation by spherical
particles (Bohren and Huffman, 1998).
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Light scattering by particles - Mie approximation
For more complex shapes such as
coated spheres, multispheres,
spheroids, and infinite cylinders there
are extensions which express the
solution in terms of infinite series.
1
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Light scattering by particles - Mie approximation
1
There are codes available to study light
scattering in Mie approximation for Codes
for electromagnetic scattering by
spheres|spheres, layered spheres, and
multiple spheres and Codes for
electromagnetic scattering by
cylinders|cylinders.
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Light scattering by particles - Discrete dipole approximation
There are several techniques for
computing scattering of radiation by
particles of arbitrary shape. The
discrete dipole approximation is an
approximation of the continuum target
by a finite array of polarizable points.
The points acquire dipole moments in
response to the local electric field.
The dipoles of these points interact
with one another via their electric
1
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Light scattering by particles - Discrete dipole approximation
There are
Discrete_dipole_approximation_codes|
DDA codes available to calculate light
scattering properties in DDA
approximation.
1
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Dynamic Light Scattering
1
[http://books.google.com/books?id=vBB54ABhmuE
Cprintsec=frontcover Dynamic Light Scattering]
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Dynamic Light Scattering - Description
1
If the light source is a laser, and thus
is monochrome|monochromatic and
Coherence (physics)|coherent, the
scattering intensity fluctuates over
time
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Dynamic Light Scattering - Description
1
The dynamic information of the particles is
derived from an autocorrelation of the
intensity trace recorded during the
experiment. The second order
autocorrelation curve is generated from
the intensity trace as follows:
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Dynamic Light Scattering - Description
where g^2(q;\tau) is the
autocorrelation function at a
particular wave vector, q, and delay
time, \tau, and I is the intensity. The
angular brackets denote the expected
value operator, which in some texts is
denoted by a capital E.
1
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Dynamic Light Scattering - Description
1
At short time delays, the correlation is high
because the particles do not have a
chance to move to a great extent from the
initial state that they were in
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Dynamic Light Scattering - Description
1
where the parameter 'β' is a correction
factor that depends on the geometry
and alignment of the laser beam in the
light scattering setup. It is roughly
equal to the inverse of the number of
speckle (see Speckle pattern) from
which light is collected. A smaller
focus of the laser beam yields a
coarser speckle pattern, a lower
number of speckle on the detector,
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Dynamic Light Scattering - Description
1
The most important use of the autocorrelation
function is its use for size determination.
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Dynamic Light Scattering - Multiple scattering
1
Alternatively, in the limit of strong
multiple scattering, a variant of
dynamic light scattering called
diffusing-wave spectroscopy can be
applied.
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Dynamic Light Scattering - Introduction
1
Once the autocorrelation data have
been generated, different
mathematical approaches can be
employed to determine 'information'
from it. Analysis of the scattering is
facilitated when particles do not
interact through collisions or
electrostatic forces between ions.
Particle-particle collisions can be
suppressed by dilution, and charge
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Dynamic Light Scattering - Introduction
1
The simplest approach is to treat the
first order autocorrelation function as a
single exponential decay. This is
appropriate for a monodisperse
population.
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Dynamic Light Scattering - Introduction
where Γ is the decay rate. The
translational diffusion coefficient Dt
may be derived at a single angle or at
a range of angles depending on the
wave vector q.
1
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Dynamic Light Scattering - Introduction
At certain angles the scattering
intensity of some particles will
completely overwhelm the weak
scattering signal of other particles,
thus making them invisible to the data
analysis at this angle
1
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Dynamic Light Scattering - Introduction
1
So, for example, colloidal gold with a layer
of surfactant will appear larger by dynamic
light scattering (which includes the
surfactant layer) than by transmission
electron microscopy (which does not see
the layer due to poor contrast)
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Dynamic Light Scattering - Introduction
In most cases, samples are
polydisperse. Thus, the
autocorrelation function is a sum of the
exponential decays corresponding to
each of the species in the population.
1
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Dynamic Light Scattering - Introduction
It is tempting to obtain data for
g^1(q;\tau) and attempt to invert the
above to extract G(Γ). Since G(Γ) is
proportional to the relative scattering
from each species, it contains
information on the distribution of
sizes. However, this is known as an
ill-posed problem. The methods
described below (and others) have
been developed to extract as much
useful information as possible from
an autocorrelation function.
1
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Dynamic Light Scattering - Cumulant method
1
One of the most common methods is the
cumulant method, from which in addition
to the sum of the exponentials above,
more information can be derived about the
variance of the system as follows:
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Dynamic Light Scattering - Cumulant method
1
: \ g^1(q;\tau) = \exp\left(-\bar\tau\right)
\left(1 + \frac\tau^2 - \frac\tau^3 +
\cdots\right)
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Dynamic Light Scattering - Cumulant method
where \scriptstyle \bar is the average
decay rate and \scriptstyle \mu_2/\bar^2
is the second order polydispersity
index (or an indication of the variance).
A third-order polydispersity index may
also be derived but this is necessary
only if the particles of the system are
highly polydisperse. The z-averaged
translational diffusion coefficient Dz
may be derived at a single angle or at a
1
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Dynamic Light Scattering - Cumulant method
1
One must note that the cumulant method
is valid for small \ \tau and sufficiently
narrow G(Γ). One should seldom use
parameters beyond µ3, because overfitting
data with many parameters in a powerseries expansion will render all the
parameters including \scriptstyle \bar and
µ2, less precise.
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Dynamic Light Scattering - Cumulant method
1
The cumulant method is far less affected by
experimental noise than the methods below.
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Dynamic Light Scattering - CONTIN algorithm
1
An alternative method for analyzing the
autocorrelation function can be achieved
through an inverse Laplace transform
known as CONTIN developed by Steven
Provencher
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Dynamic Light Scattering - Maximum entropy method
The Principle of maximum
entropy|Maximum entropy method is an
analysis method that has great
developmental potential. The method is
also used for the quantification of
sedimentation velocity data from
analytical ultracentrifugation. The
maximum entropy method involves a
number of iterative steps to minimize
the deviation of the fitted data from the
experimental data and subsequently
reduce the χ2 of the fitted data.
1
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Dynamic Light Scattering - Applications
1
DLS is used to characterize size of various
particles including proteins, polymers,
micelles, carbohydrates, and
nanoparticles. If the system is
monodisperse, the mean effective
diameter of the particles can be
determined. This measurement depends
on the size of the particle core, the size of
surface structures, particle concentration,
and the type of ions in the medium.
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Dynamic Light Scattering - Applications
1
For more than two populations CONTIN analysis at
several scattering angles is required
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Dynamic Light Scattering - Applications
Stability studies can be done
conveniently using DLS. Periodical DLS
measurements of a sample can show
whether the particles aggregate over
time by seeing whether the
hydrodynamic radius of the particle
increases. If particles aggregate, there
will be a larger population of particles
with a larger radius. Additionally, in
certain DLS machines, stability
depending on temperature can be
analyzed by controlling the temperature
1
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Backscattering
In physics, 'backscatter' (or
'backscattering') is the reflection of waves,
particles, or signals back to the direction
from which they came. It is a diffuse
reflection due to scattering, as opposed to
specular reflection like a mirror.
Backscattering has important applications
in astronomy, photography and medical
ultrasonography.
1
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Backscattering - Backscatter of waves in physical space
1
Backscattering can occur in quite different
physical situations, where the incoming
waves or particles are deflected from their
original direction by different mechanisms:
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Backscattering - Backscatter of waves in physical space
1
*Diffuse reflection from large particles and
Mie scattering, causing alpenglow and
gegenschein, and showing up in weather
radar;
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Backscattering - Backscatter of waves in physical space
*inelastic collisions between
electromagnetic waves and the
transmitting medium (Brillouin
scattering and Raman scattering),
important in fiber optics, see below;
1
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Backscattering - Backscatter of waves in physical space
1
*elastic collisions between accelerated ions
and a sample (Rutherford backscattering)
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Backscattering - Backscatter of waves in physical space
1
*Bragg's law|Bragg diffraction from
crystals, used in inelastic scattering
experiments (neutron backscattering,
X-ray backscattering spectroscopy);
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Backscattering - Backscatter of waves in physical space
1
*Compton scattering, used in
Backscatter X-ray imaging.
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Backscattering - Backscatter of waves in physical space
1
Sometimes, the scattering is more or less
isotropic, i. e. the incoming particles are
scattered randomly in various directions,
with no particular preference for backward
scattering. In these cases, the term
backscattering just designates the detector
location chosen for some practical
reasons:
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Backscattering - Backscatter of waves in physical space
1
*in X-ray imaging, backscattering means just
the opposite of transmission imaging;
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Backscattering - Backscatter of waves in physical space
1
*in inelastic neutron or X-ray spectroscopy,
backscattering geometry is chosen
because it optimizes the energy resolution;
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Backscattering - Backscatter of waves in physical space
1
*In alpenglow, red light prevails because
the blue part of the spectrum is depleted
by Rayleigh scattering.
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Backscattering - Backscatter of waves in physical space
1
*In gegenschein, constructive interference might
play a role (this needs verification).
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Backscattering - Backscatter of waves in physical space
1
*Coherent backscattering is observed
in random media; for visible light most
typically in Suspension
(chemistry)|suspensions like milk. Due
to weak localization, enhanced multiple
scattering is observed in back
direction.
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Backscattering - Backscatter of waves in physical space
1
** The Back Scattering Alignment (BSA) coordinate
system is often used in radar applications
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Backscattering - Backscatter of waves in physical space
** The Forward Scattering Alignment
(FSA) coordinate system is primarily used
in optical applications
1
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Backscattering - Backscatter in photography
The term backscatter in photography
refers to light from a Flash
(photography)|flash or strobe reflecting
back from particles in the lens's field of
view causing specks of light to appear in
the photo
1
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Coherent backscattering
1
In physics, 'coherent backscattering'
is observed when coherence
(physics)|coherent radiation (such as
a laser beam) propagates through a
medium which has a large number of
scattering centers (such as milk or a
thick cloud) of size comparable to the
wavelength of the radiation.
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Coherent backscattering
1
The waves are scattered many times while
traveling through the medium. Even for
incoherent radiation, the scattering
typically reaches a maxima and
minima|local maximum in the direction of
backscattering. For coherent radiation,
however, the peak is two times higher.
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Coherent backscattering
1
Coherent backscattering is very difficult to
detect and measure for two reasons
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Coherent backscattering
This substantial
enhancement is called the
cone of coherent
backscattering.
1
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Coherent backscattering
The enhanced backscattering relies on the
constructive interference between reverse paths
1
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Mott scattering
1
'Mott scattering', also referred to as
spin-coupling inelastic Rutherford
scattering|Coulomb scattering, is the
separation of the two spin states of an
electron beam by scattering the beam
off the Coulomb field of heavy atoms.
It is mostly used to measure the spin
polarization of an electron beam.
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Mott scattering
In lay terms, Mott scattering is
similar to Rutherford
Scattering|Rutherford scattering but
electrons are used instead of a Alpha
particle|lpha particles as they do not
interact via the strong force (only
weak and electromagnetic). This
enables them to penetrate the atomic
nucleus, giving valuable insight into
the nuclear structure.
1
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Mott scattering
The electrons are often fired at gold foil
because gold has a high atomic number
(Z), is non-reactive (does not form an
oxide layer), and can be easily made into
a thin film (reducing multiple scattering).
The presence of a spin-orbit term in the
scattering potential introduces a spin
dependence in the scattering cross
section. Two detectors at exactly the same
scattering angle to the left and right of the
1
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Mott scattering
is proportional to the degree of spin
polarization P according to A = SP, where
S is the Sherman function.
1
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Mott scattering
1
The 'Mott cross section formula' is the
mathematical description of the
scattering of a high energy electron
beam from an atomic nucleus-sized
positively charged point in space. The
Mott scattering is the theoretical
diffraction pattern produced by such a
mathematical model. It is used as the
beginning point in calculations in
electron scattering diffraction studies.
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Mott scattering
The equation for the Mott cross section
includes an inelastic scattering term to
take into account the recoil of the target
proton or nucleus. It also can be corrected
for relativistic effects of high energy
electrons, and for their magnetic
moment.[http://hyperphysics.phyastr.gsu.edu/hbase/nuclear/elescat.html
Hyperphysics]
1
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Mott scattering
When an experimentally found
diffraction pattern deviates from the
mathematically derived Mott
scattering, it gives clues as to the size
and shape of an atomic nucleusSee:
ME Rose 1948 The Charge
Distribution in Nuclei and the
Scattering of High Energy Electrons
Physical Review '73' #4 p279-84; Also
the Hyperphysics reference preceding
1
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Mott scattering
The Born approximation of the
diffraction of a beam of electrons by
atomic nuclei is an extension of Mott
scattering.See: NF Mott and HSW
Massey 1965 The Theory of Atomic
Collisions, Third Edition (Oxford:
Oxford University Press)
1
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Leonid Mandelshtam - Discovery of the combinatorial scattering of light
Mandelstam, New phenomenon in
scattering of light (preliminary report),
Journal of the Russian Physico-Chemical
Society, Physics Section '60', 335 (1928)
1
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Leonid Mandelshtam - Discovery of the combinatorial scattering of light
1
Thus, combinatorial scattering of light was
discovered by Mandelstam and Landsberg
a week earlier than by Raman and
Krishnan
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Elastic scattering
1
During elastic scattering of high-energy
subatomic particles, linear energy transfer
(LET) takes place until the incident
particle's energy and speed has been
reduced to the same as its surroundings,
at which point the particle is stopped.
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Elastic scattering - Electron elastic scattering
1
In many electron diffraction techniques like
reflection high energy electron diffraction
(RHEED), transmission electron diffraction
(TED), and gas electron diffraction (GED),
where the incident electrons have sufficiently
high energy ( 10 keV), the elastic electron
scattering becomes the main component of
the scattering process and the scattering
intensity is expressed as a function of the
momentum transfer defined as the difference
between the momentum vector of the incident
electron and that of the scattered electron.
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Elastic scattering - Optical elastic scattering
* In Rayleigh scattering a photon
penetrates into a medium composed
of particles whose sizes are much
smaller than the wavelength of the
incident photon. In this scattering
process, the energy (and therefore the
wavelength) of the incident photon is
conserved and only its direction is
changed. In this case, the scattering
intensity is proportional to the fourth
power of the reciprocal wavelength of
the incident photon.
1
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Elastic scattering - Nuclear particle physics
1
In nuclear reactors, the neutron's
mean free path is critical as it
undergoes elastic scattering on its
way to becoming a slow-moving
thermal neutron.
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Elastic scattering - Nuclear particle physics
Besides elastic scattering, charged
particles also undergo effects from their
elementary charge, which repels them
away from nuclei and causes their path
to be curved inside an electric field.
Particles can also undergo inelastic
scattering and capture due to nuclear
reactions. Protons and neutrons do this
more often than heavier particles.
Neutrons are also capable of causing
Nuclear fission|fission in an incident
nucleus. Light nuclei like deuterium and
lithium can combine in nuclear fusion.
1
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Bhabha scattering
In quantum electrodynamics, 'Bhabha
scattering' is the electron-positron scattering
process:
1
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Bhabha scattering
1
There are two leading-order Feynman
diagrams contributing to this
interaction: an annihilation process
and a scattering process. Bhabha
scattering is named after the Indian
physicist Homi J. Bhabha.
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Bhabha scattering
1
The Bhabha scattering rate is used as a
luminosity monitor in electron-positron
colliders.
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Bhabha scattering - Differential cross section
|Where we use:\gamma^\mu \, are the
Gamma matrices,u, \ \mathrm \ \bar\, are
the four-component spinors for fermions,
whilev, \ \mathrm \ \bar\, are the fourcomponent spinors for anti-fermions (see
Dirac equation#Four spinor|Four spinors).
1
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Bhabha scattering - Magnitude squared of M
|align=center |
(complex conjugate will
flip order)
1
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Bhabha scattering - Magnitude squared of M
1
|align=center | (move terms that depend on same
momentum to be next to each other)
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Bhabha scattering - Sum over spins
1
Next, we'd like to sum over spins of all four
particles. Let s and s' be the spin of the
electron and r and r' be the spin of the
positron.
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Bhabha scattering - Sum over spins
1
:: \bar_ \gamma^\mu (\sum_v_ \bar_)
\gamma^\nu v_ \right) \left(\sum_ \bar_
\gamma_\mu (\sum_\left( \Big(\sum_ v_
\bar_ \Big) \gamma^\mu \Big(\sum_v_
\bar_ \Big) \gamma^\nu \right)
\operatorname \left( \Big(\sum_ u_p
\bar_ \Big) \gamma_\mu \Big( \sum_ +
k'^\nu k^\mu \right) + 4 m^2 \eta^
\right) \left( 4 \left( _\mu p_\nu - (p' \cdot
p)\eta_ + p'_\nu p_\mu \right) + 4 m^2
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Bhabha scattering - Sum over spins
1
Now that is the exact form, in the case
of electrons one is usually interested
in energy scales that far exceed the
electron mass. Neglecting the
electron mass yields the simplified
form:
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Bhabha scattering - Annihilation term (s-channel)
The process for finding the annihilation
term is similar to the above. Since the two
diagrams are related by crossing
symmetry, and the initial and final state
particles are the same, it is sufficient to
permute the momenta, yielding
1
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Bhabha scattering - Solution
1
Evaluating the interference term along
the same lines and adding the three
terms yields the final result
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Bhabha scattering - Completeness relations
The completeness relations for the
Dirac_spinor#Four-spinor_for_particles|four-spinors
u and v are
1
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Bhabha scattering - Completeness relations
::\sum__p \bar^_ps=1,2v^_p
\bar^_p\dagger\rho\mu\sigma\nu\rho\sigma\rho\nu\
mu\sigma|
1
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Bhabha scattering - Completeness relations
1
|\operatorname\left( (p\!\!\!/' + m) \gamma_\mu
(p\!\!\!/ + m) \gamma_\nu \right) \,
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Bhabha scattering - Completeness relations
1
| = \operatorname\left( p\!\!\!/' \gamma_\mu
p\!\!\!/ \gamma_\nu \right) +
\operatorname\left(m \gamma_\mu p\!\!\!/
\gamma_\nu \right) \,
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Bhabha scattering - Completeness relations
| + \operatorname\left( p\!\!\!/'
\gamma_\mu m \gamma_\nu \right) +
\operatorname\left(m^2 \gamma_\mu
\gamma_\nu \right) \,
1
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Bhabha scattering - Completeness relations
1
|align=center|(the two middle terms
are zero because of (1))
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Bhabha scattering - Completeness relations
|align=center|(use
identity (2) for the term
on the right)
1
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Bhabha scattering - Completeness relations
|= ^ p^\sigma \operatorname\left(
\gamma_\rho \gamma_\mu
\gamma_\sigma \gamma_\nu \right) + m^2
\cdot 4\eta_ \,
1
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Bhabha scattering - Completeness relations
1
|align=center|(now use identity
(3) for the term on the left)
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Bhabha scattering - Uses
1
Bhabha scattering has been used as a
luminosity monitor in a number of e+e−
collider physics experiments. The
accurate measurement of luminosity is
necessary for accurate measurements of
cross sections.
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Bhabha scattering - Uses
1
* Small-angle Bhabha scattering was used
to measure the luminosity of the 1993 run
of the Stanford Large Detector (SLD), with
a relative uncertainty of less than
0.5%.[http://adsabs.harvard.edu/abs/1995
PhDT.......160W A Study of Small Angle
Radiative Bhabha Scattering and
Measurement of the Luminosity at SLD]
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Bhabha scattering - Uses
1
* Electron-positron colliders operating in
the region of the low-lying hadronic
resonances (about 1 GeV to 10 GeV),
such as the Beijing Electron Synchrotron
(BES) and the Belle experiment|Belle and
BaBar experiment|BaBar B-factory
experiments, use large-angle Bhabha
scattering as a luminosity monitor
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Incoherent scattering
'Incoherent scattering' is a type of
scattering phenomenon in physics.
The term is most commonly used
when referring to the scattering of an
electromagnetic wave (usually light or
radio frequency) by random
fluctuations in a gas of particles
(most often electrons).
1
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Incoherent scattering
1
A radar beam scattering off electrons in
the ionospheric Plasma (physics)|plasma
creates an incoherent scatter return
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Brillouin scattering
'Brillouin scattering', named after Léon
Brillouin, occurs when light, transmitted by
a transparent carrier interacts with that
carrier's time--space-periodic variations in
refractive index. As described by optics,
the index of refraction of a transparent
material changes under deformation
(compression-distension or shearskewing).
1
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Brillouin scattering
1
The result of the interaction between the
light-wave and the carrier-deformationwave is that a fraction from the passingthrough light-wave changes its momentum
(thus its frequency and energy) along
preferential angles, as if by being
diffracted by an oscillating 3-D grating.
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Brillouin scattering
If the involved light carrier is a solid
crystal, a macromolecular chain
condensate or a viscous liquid, then the
low frequency atomic-chain-deformation
waves in the carrier (represented as a
quasiparticle) could be for example: 1.
mass oscillation (acoustic) modes (called
phonons); 2. charge displacement modes
(in dielectrics, called polarons); 3.
magnetic spin oscillation modes (in
magnetic materials, called magnons).
1
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Brillouin scattering - Mechanism
1
Thus the Brillouin scattering can be used
to measure the energies, wavelengths and
frequencies of various atomic chain
oscillation types ('quasiparticles')
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Brillouin scattering - Contrast with Rayleigh scattering
Rayleigh scattering, too, can be
considered to be due to fluctuation in the
density, composition and orientation of
molecules, and hence of refraction index,
in small volumes of matter (particularly in
gases or liquids). The difference is that
Rayleigh scattering considers only random
and incoherent thermal fluctuations, in
contrast with the correlated, periodic
fluctuations (phonons) that cause the
Brillouin scattering.
1
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Brillouin scattering - Contrast with Raman scattering
Experimentally, the frequency shifts in
Brillouin scattering are detected with an
interferometer, while Raman setup can be
based on either interferometer or
dispersive (grating) spectrometer.
1
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Brillouin scattering - Stimulated Brillouin scattering
Stimulated Brillouin scattering is one
effect by which Nonlinear
optics#Optical_phase_conjugation|optical
phase conjugation can take place.
1
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Brillouin scattering - Discovery
Inelastic scattering of light by acoustic
phonons was first predicted by Léon
Brillouin in 1922. Leonid Mandelstam is
believed to have recognised the possibility
of such scattering as early as 1918, but he
published it only in 1926.Feînberg, E.L.:
1
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Brillouin scattering - Discovery
1
The forefather, Uspekhi Fizicheskikh Nauk,
Vol. '172', 2002 (Physics-Uspekhi, '45', 81 (2002)
)
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Brillouin scattering - Discovery
1
In order to credit Mandelstam the effect
is also called Brillouin-Mandelstam
scattering (BMS). Other commonly
used names are Brillouin light
scattering (BLS) and BrillouinMandelstam light scattering (BMLS).
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Brillouin scattering - Discovery
1
The process of stimulated Brillouin
scattering (SBS) was first observed by
Chiao et al. in 1964. The optical phase
conjugation aspect of the SBS process
was discovered by Zel’dovich et al. in
1972.
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Brillouin scattering - Fiber Optic Sensing
1
Brillouin scattering can also be employed
to sense Deformation
(mechanics)|mechanical strain and
temperature in optical fibers.
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Inelastic scattering
In general, scattering due to inelastic
collisions will be inelastic, but, since elastic
collisions often transfer kinetic energy
between particles, scattering due to elastic
collisions can also be inelastic, as in
Compton scattering (see below).
1
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Inelastic scattering - Electrons
1
Deep inelastic scattering of electrons
from protons provided the first direct
evidence for the existence of quarks.
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Inelastic scattering - Photons
1
The blue shift can be observed when
internal energy of the matter is
transferred to the photon; this process
is called anti-Stokes Raman
scattering.
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Inelastic scattering - Photons
Inelastic scattering is seen in the
interaction between an electron and a
photon. When a high-energy photon
collides with a free electron and
transfers energy, the process is called
Compton scattering. Furthermore,
when an electron with relativistic
energy collides with an infrared or
visible photon, the electron gives
energy to the photon; this process is
1
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Inelastic scattering - Neutrons
1
In inelastic scattering, neutrons are readily
absorbed in a process called neutron
capture and attributes to the neutron
activation of the nucleus
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Inelastic scattering - Molecular collisions
Inelastic scattering is common in
molecular collisions. Any collision which
leads to a chemical reaction will be
inelastic, but the term inelastic scattering
is reserved for those collisions which do
not result in reactions. There is a transfer
of energy between the translational mode
(kinetic energy) and rotational and
vibrational modes.
1
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Inelastic scattering - Molecular collisions
If the transferred energy is small
compared to the incident energy of the
scattered particle, one speaks of
quasielastic scattering.
1
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Scattering theory
The inverse scattering problem 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.
1
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Scattering theory
Since its early statement for
radar|radiolocation, the problem has
found vast number of applications,
such as acoustic location|echolocation,
geophysical survey, nondestructive
testing, medical imaging and quantum
field theory, to name just a few.
1
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Scattering theory - Conceptual underpinnings
1
The concepts used in
scattering theory go
by different names in
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Scattering theory - 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
1
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Scattering theory - Composite targets and range equations
1
The above ordinary first-order differential equation
has solutions of the form:
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Scattering theory - Composite targets and range equations
1
: I = I_o e^ = I_o e^ =
I_o e^ \tau ,
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Scattering theory - Composite targets and range equations
where Io is the initial flux, path length
Δx≡xminus;xo, the second equality defines
an interaction mean free path λ, the third
uses the number of targets per unit
volume η to define an area cross section
(physics)|cross-section 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/lambda; =eta;sigma; =rho;/tau;, as
1
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Scattering theory - Composite targets and range equations
1
In electromagnetic absorption spectroscopy,
for example, interaction coefficient (e.g
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Scattering theory - In theoretical physics
In the case of classical
electrodynamics, the differential
equation is again the wave equation,
and the scattering of light or radio
waves is studied
1
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Scattering theory - In theoretical physics
1
There are two predominant techniques of
finding solutions to scattering problems:
partial wave analysis, and the Born
approximation.
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Scattering theory - Elastic and inelastic scattering
1
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 a scattering atom, or the
complete annihilation of a scattering
particle and the creation of entirely new
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Scattering theory - Elastic and inelastic scattering
1
The scattering of two hydrogen atoms
will disturb the state of each atom,
resulting in one or both becoming
excited, or even ionization|ionized,
representing an inelastic scattering
process.
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Scattering theory - The mathematical framework
1
The scattering matrix then pairs solutions in the
distant past to those in the distant future.
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Scattering theory - The mathematical framework
1
The study of inelastic scattering then
asks how discrete and continuous
spectra are mixed together.
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Scattering theory - The mathematical framework
1
An important, notable development is the
inverse scattering transform, central to the
solution of many exactly solvable models.
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Møller scattering
1
Nevertheless Møller scattering remains a
paradigmatic process within the theory of
particle interactions.
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Møller scattering
1
We can express this process in the usual
notation, often used in particle physics:
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Møller scattering
1
In quantum electrodynamics, there are two
tree-level Feynman diagrams describing
the process: a Mandelstam variables|tchannel diagram in which the electrons
exchange a photon and a similar uchannel diagram. Crossing symmetry, one
of the tricks often used to evaluate
Feynman diagrams, in this case implies
that Møller scattering should have the
same cross section as Bhabha scattering
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Møller scattering
The asymmetry in
Møller scattering is
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Møller scattering
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A_=-m E \frac \pi \alpha 16 \sin^2 \Theta_ \textrm
cmEmpty section|date=July 2010
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Chaotic scattering
In a chaotic scattering system, a
minute change in the impact
parameter, may give rise to a very
large change in the exit parameters.
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Chaotic scattering - Gaspard–Rice system
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—also known simply as the three-disc
system—which embodies many of the
important concepts in chaotic
scattering while being simple and
easy to understand and simulate
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Chaotic scattering - Gaspard–Rice system
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Figure 1 illustrates this system while Figure 2
shows two example trajectories
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Chaotic scattering - Decay rate
If we introduce a large number of
particles with uniformly distributed
impact parameters, the rate at which
they exit the system is known as the
decay rate
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Chaotic scattering - Decay rate
We expect the
number of particles
remaining in the
system, N(T), to vary
as:
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Chaotic scattering - Decay rate
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where n is the total
number of particles.
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Chaotic scattering - Decay rate
Figure 3 shows a plot of the pathlength versus the number of particles
for a simulation of one million (1e6)
particles started with random impact
parameter, b. A fitted straight line of
negative slope, \gamma=0.739 is
overlaid. The path-length, s, is
equivalent to the decay time, T,
provided we scale the (constant) speed
appropriately.
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Chaotic scattering - An experimental system and the stable manifold
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Figure 4 shows an
experimental
realization of the
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Chaotic scattering - An experimental system and the stable manifold
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Gaspard–Rice system using
a laser instead of a point
particle.
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Chaotic scattering - An experimental system and the stable manifold
As anyone who's actually
tried this knows, this is not a
very effective
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Chaotic scattering - An experimental system and the stable manifold
a more effective
method is to direct
coloured light
through the gaps
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Chaotic scattering - An experimental system and the stable manifold
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between the discs (or in this case, tape
coloured strips of paper across pairs of
cylinders)
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Chaotic scattering - An experimental system and the stable manifold
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and view the reflections
through an open gap.
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Chaotic scattering - An experimental system and the stable manifold
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The result is a complex
pattern of stripes of
alternating colour, as
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Chaotic scattering - An experimental system and the stable manifold
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shown below, seen more clearly
in the simulated version below
that.
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Chaotic scattering - An experimental system and the stable manifold
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Figures 5 and 6 show the basins of
attraction for each
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Chaotic scattering - An experimental system and the stable manifold
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impact parameter, b, that is,
for a given value of b,
through which gap
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Chaotic scattering - An experimental system and the stable manifold
does the particle
exit? The basin
boundaries form a
Cantor set and
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Chaotic scattering - The invariant set and the symbolic dynamics
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So long as it is symmetric, we can easily
think of the system as an iterated function
map, a common method of representing a
chaotic, dynamical system.
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Chaotic scattering - The invariant set and the symbolic dynamics
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Figure 7 shows one possible representation of the
variables, with the first variable,
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Chaotic scattering - The invariant set and the symbolic dynamics
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\theta \in [-\pi, \pi], representing the
angle around the disc at rebound and
the second, \phi \in [-\pi/2, \pi/2],
representing the impact/rebound
angle relative to the disc.
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Chaotic scattering - The invariant set and the symbolic dynamics
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A subset of these two variables, called the invariant
set will map onto themselves.
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Chaotic scattering - The invariant set and the symbolic dynamics
This set, four members of which are
shown in Figures 8 and 9, will be fractal,
totally non-attracting
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Chaotic scattering - The invariant set and the symbolic dynamics
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and of measure (mathematics)|measure
zero. This is an interesting inversion of
the more normally discussed chaotic
systems in which the fractal invariant set is
attracting and in fact comprises the
basin[s] of attraction. Note that the totally
non-attracting nature of the invariant set is
another property of a hyperbolic chaotic
scatterer.
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Chaotic scattering - The invariant set and the symbolic dynamics
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Each member of the invariant set can be
modelled using symbolic dynamics: the
trajectory is labelled based on each of the
discs off of which it rebounds.
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Chaotic scattering - The invariant set and the symbolic dynamics
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For the four members shown in Figures 8 and 9,
the symbolic dynamics will be as follows:
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Chaotic scattering - The invariant set and the symbolic dynamics
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Members of the stable manifold may be
likewise represented, except each
sequence will have a starting point. When
you consider that a member of the
invariant set must fit in the boundaries
between two basins of attraction, it is
apparent that, if perturbed, the trajectory
may exit anywhere along the sequence.
Thus it should also be apparent that an
infinite number of alternating basins of all
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Chaotic scattering - The invariant set and the symbolic dynamics
Because of their unstable nature, it is
difficult to access members of the invariant
set or the stable manifold directly. The
uncertainty exponent is ideally tailored to
measure the fractal dimension of this type
of system. Once again using the single
impact parameter, b, we perform multiple
trials with random impact parameters,
perturbing them by a minute amount,
\epsilon, and counting how frequently the
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Chaotic scattering - The invariant set and the symbolic dynamics
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Note that even though the system is two
dimensional, a single impact parameter is
sufficient to measure the fractal dimension
of the stable manifold. This is
demonstrated in Figure 10, which shows
the basins of attraction plotted as a
function of a dual impact parameter, \theta
and \phi. The stable manifold, which can
be seen in the boundaries between the
basins, is fractal along only one
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Chaotic scattering - The invariant set and the symbolic dynamics
Figure 11 plots the uncertainty
fraction, f, as a function of the
uncertainty, \epsilon for a simulated
Gaspard–Rice system. The slope of
the fitted curve returns the
uncertainty exponent, \gamma=0.380,
thus the box-counting dimension of
the stable manifold is, D_0=N\gamma=2-0.380=1.62. The invariant
set is the intersection of the stable and
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Chaotic scattering - The invariant set and the symbolic dynamics
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Since the system is the same whether run
forwards or backwards, the unstable
manifold is simply the mirror image of the
stable manifold and their fractal
dimensions will be equal.
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Chaotic scattering - The invariant set and the symbolic dynamics
On this basis we can
calculate the fractal
dimension of the invariant
set:
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Chaotic scattering - The invariant set and the symbolic dynamics
where D_s and D_u are the fractal
dimensions of the stable and unstable
manifolds, respectively and N=2 is the
dimensionality of the system. The
fractal dimension of the invariant set
is D=1.24.
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Chaotic scattering - Relationship between the fractal dimension, decay rate and
Lyapunov exponents
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Thus we can see that both will affect
the decay rate as captured in the
following conjecture for a twodimensional scattering system:
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Chaotic scattering - Relationship between the fractal dimension, decay rate and
Lyapunov exponents
where D1 is the information
dimension and h1 and h2 are the small
and large Lyapunov exponents,
respectively. For an attractor,
\gamma=\infty and it reduces to the
Kaplan–Yorke conjecture.
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Raman in extensive theoretical and experimental studies on light scattering,
molecular optics and in the discovery of the Raman Effect (1928). More
recently has been publishing many valuable investigations (Phil Trans Royal
Society and elsewhere) on the significance of magnetic anisotropy in relation to
crystal architecture and thermo-magnetic behaviour at the lowest temperatures.
Has published important work on pleochroism in crystals and its relation to
photo-dissociation. Leader of an active school of research in Calcutta.
http://www2.royalsociety.org/DServe/dserve.exe?dsqIniDserve.inidsqAppArchiv
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edsqDbCatalogdsqSearchRefNo
In 1942, he moved to Allahabad University
as Professor and Head of the Department
of Physics where he took up the physics of
solids, in particular of metals.
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Raman in extensive theoretical and experimental studies on light scattering,
molecular optics and in the discovery of the Raman Effect (1928). More
recently has been publishing many valuable investigations (Phil Trans Royal
Society and elsewhere) on the significance of magnetic anisotropy in relation to
crystal architecture and thermo-magnetic behaviour at the lowest temperatures.
Has published important work on pleochroism in crystals and its relation to
photo-dissociation. Leader of an active school of research in Calcutta.
http://www2.royalsociety.org/DServe/dserve.exe?dsqIniDserve.inidsqAppArchiv
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edsqDbCatalogdsqSearchRefNo
He was knighted in the 1946 Birthday
Honours List[http://www.londongazette.co.uk/issues/37598/supplements/2
757 London Gazette, 4 June 1946] and
awarded the Padma Bhushan by the
Government of India in
1954.http://www.mha.nic.in/pdfs/PadmaAw
ards1954-2007.pdf He was the first
recipient of the prestigious Bhatnagar
Award in 1958.
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Coherent scattering - Single and multiple scattering
In certain rare circumstances,
multiple scattering may only involve a
small number of interactions such
that the randomness is not completely
averaged out
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Resonance Raman spectroscopy - Raman Scattering
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This type of scattering is known as
Rayleigh scattering
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Resonance Raman spectroscopy - Raman Scattering
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However, it is possible for the molecules to
relax back to a vibrational state that is
higher in energy than the state they
started in
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Resonance Raman spectroscopy - Raman Scattering
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Since the Stokes lines and anti-Stokes
lines gain and lose the same amount
of energy, they are symmetric with
respect to the peak due to elastic
(Rayleigh) scattering (\Delta \bar
\nu=0)
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Cutting Crew - 1988–1990: The Scattering
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Although a video for the title track did air
briefly in the UK and North America, The
Scattering failed to chart.
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Cutting Crew - The Scattering (1989)
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The Scattering will probably seem
dated to anyone who isn't an '80s
enthusiast, but it's tasty nostalgia for
people who remember the decade
fondly
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Slit lamp - Scattering sclero-corneal illumination
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With this type of illumination, a wide light
beam is directed onto the limbal region of
the cornea at an extremely low angle of
incidence and with a laterally de-centered
illuminating prism
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Sudbury Neutrino Observatory - Electron elastic scattering
In the elastic scattering interaction, a
neutrino collides with an atomic electron
and imparts some of its energy to the
electron
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Bug zapper - Scattering
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Research has shown that when insects
are electric shock|electrocuted bug
zappers can spread a mist containing
insect parts up to about from the device
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Glass coloring and color marking - Color caused by scattering
Glass containing two or more phase
(matter)|phases with different refractive
index|refractive indices shows coloring
based on the Tyndall effect and
explained by the Mie theory, if the
dimensions of the phases are similar or
larger than the
Frequency#Physics_of_light|wavelengt
h of visible light. The scattered light is
blue and violet as seen in the image,
while the transmitted light is yellow and
red.
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