Scattering cross-section
Encyclopedia
The scattering cross-section, σscat, is a hypothetical area which describes the likelihood of light (or other radiation) being scattered
by a particle. In general, the scattering cross-section is different from the geometrical cross-section of a particle, and it depends upon the wavelength
of light and the permittivity
, shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present. In terms of area, the total cross-section (σ) is the sum of the cross-sections due to absorption
, scattering
and luminescence
The total cross-section is related to the absorbance of the light intensity through Beer-Lambert's law, which says absorbance is proportional to concentration:
, where C is the concentration as a number density, Aλ is the absorbance at a given wavelength
, λ, and l is the path length
. The extinction or absorbance of the radiation is the logarithm
(decadic or, more usually, natural
) of the reciprocal of the transmittance
:
gy weather: the droplets of water (which form the fog) scatter red light less than they scatter the shorter wavelengths present in white light, and the red rear fog light can be distinguished more clearly than the white headlights of an approaching vehicle. That is to say that the scattering cross-section of the water droplets is smaller for red light than for light of shorter wavelengths, even though the physical size of the particles is the same.
The quantity Cσscat is sometimes denoted bscat, the scattering coefficient per unit length.
.
When the scattered radiation is visible light, it is conventional to measure the path length in centimetre
s. To avoid the need for conversion factors, the scattering cross-section is expressed in cm² (1 cm² = 10−4 m²) and the number concentration in cm−3 (1 cm−3 = 10−6 m−3). The measurement of the scattering of visible light is known as nephelometry
, and is effective for particles of 2–50 µm
in diameter: as such, it is widely used in meteorology
and in the measurement of atmospheric pollution.
Scattering cross-sections are also widely used in the description of the scattering of neutron
s by atomic nuclei
. In this case, the conventional unit is the barn
, b, where 1 b = 10−28 m² = 100 fm².
The scattering of X-ray
s can also be described in terms of scattering cross-sections, in which case the square ångström
, Ų, is a convenient unit: 1 Ų = 10−20 m² = 104 pm².
, the differential cross section is defined as follows: let a beam of intensity
(measured in number of particles per area per time) be incident on a scattering center. In general, the angle at which a particle is scattered is a function of impact parameter
. The number of scattered particles per solid angle
per time,
is therefore well defined. We define the differential cross section to be
Note that this quantity has units of area. Furthermore, it depends only on the geometry of the scattering center, and not on the incident flux or distance of the detector from the scattering center. The geometric interpretation is as follows: consider particles that scatter through a solid angle
and ask what values of impact parameter produced them. These impact parameters form a differential area, 
in space. The differential cross section is simply
, the wave function of the incident particle is a plane-wave with amplitude 1, that is,
. In general the scattered wave is of the form
We then have as the definition of differential cross section
This has the simple interpretation of the probability of finding a scattered particle within a given solid angle.
The integral cross section is the integral
of the differential cross section on the whole sphere of observation (4
steradian
):
A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. Usual units are the cm2, the barn
(1 b = 10−28 m2) and the corresponding submultiples: the millibarn (1 mb = 10−3 b), the microbarn (1
b = 10−6 b), the nanobarn ( 1 nb = 10−9 b), the picobarn (1 pb = 10−12 b), and the shed (1 shed = 10−24 b). The cross section of two particles
(i.e. observed when the two particles are colliding
with each other) is a measure of the interaction event between the two particles.
The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles
) depends only on the number of incident particles per unit of time (current of incident particles 
), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction. For 
we have
es and momenta
of the colliding system are mi,
and mf, 
before and after the collision respectively, the differential cross section is given by
where the on-shell T matrix is defined by
in terms of the S matrix
. The
function is the distribution
called the Dirac delta function
. The computation of the S matrix is the main aim of the scattering theory
.
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
by a particle. In general, the scattering cross-section is different from the geometrical cross-section of a particle, and it depends upon the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of light and the permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
, shape and size of the particle. The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present. In terms of area, the total cross-section (σ) is the sum of the cross-sections due to absorption
Absorption cross section
Absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain particle-particle interaction, e.g., scattering, electromagnetic absorption, etc...
, scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
and luminescence
Luminescence
Luminescence is emission of light by a substance not resulting from heat; it is thus a form of cold body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions, or stress on a crystal. This distinguishes luminescence from incandescence, which is light emitted by a...
The total cross-section is related to the absorbance of the light intensity through Beer-Lambert's law, which says absorbance is proportional to concentration:
, where C is the concentration as a number density, Aλ is the absorbance at a given wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, λ, and l is the path length
Path length
In chemistry, the path length is defined as the distance that light travels through a sample in an analytical cell. Typically, a sample cell is made of quartz, glass, or a plastic rhombic cuvette with a volume typically ranging from 0.1 mL to 10 mL or larger used in a spectrophotometer. For the...
. The extinction or absorbance of the radiation is the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
(decadic or, more usually, natural
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
) of the reciprocal of the transmittance
Transmittance
In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, or absorption factor, which is the fraction of radiation absorbed by a sample at a specified wavelength...
:
Relation to physical size
There is no simple relationship between the scattering cross-section and the physical size of the particles, as the scattering cross-section depends on the wavelength of radiation used. This can be seen when driving in fogFog
Fog is a collection of water droplets or ice crystals suspended in the air at or near the Earth's surface. While fog is a type of stratus cloud, the term "fog" is typically distinguished from the more generic term "cloud" in that fog is low-lying, and the moisture in the fog is often generated...
gy weather: the droplets of water (which form the fog) scatter red light less than they scatter the shorter wavelengths present in white light, and the red rear fog light can be distinguished more clearly than the white headlights of an approaching vehicle. That is to say that the scattering cross-section of the water droplets is smaller for red light than for light of shorter wavelengths, even though the physical size of the particles is the same.
Meteorological range
The scattering cross-section is related to the meteorological range, LV:The quantity Cσscat is sometimes denoted bscat, the scattering coefficient per unit length.
Units
Its SI unit is the square meter, m², although smaller units are usually used in practice. The name cross-section arises because it has the dimensions of areaArea
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
.
When the scattered radiation is visible light, it is conventional to measure the path length in centimetre
Centimetre
A centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively...
s. To avoid the need for conversion factors, the scattering cross-section is expressed in cm² (1 cm² = 10−4 m²) and the number concentration in cm−3 (1 cm−3 = 10−6 m−3). The measurement of the scattering of visible light is known as nephelometry
Nephelometry
Nephelometry is a technique used in immunology to determine levels of IgM, IgG, and IgA.It is performed by measuring the turbidity in a water sample by passing light through the sample being measured...
, and is effective for particles of 2–50 µm
Micrometre
A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...
in diameter: as such, it is widely used in meteorology
Meteorology
Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
and in the measurement of atmospheric pollution.
Scattering cross-sections are also widely used in the description of the scattering of neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s by atomic nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
. In this case, the conventional unit is the barn
Barn (unit)
A barn is a unit of area. Originally used in nuclear physics for expressing the cross sectional area of nuclei and nuclear reactions, today it is used in all fields of high energy physics to express the cross sections of any scattering process, and is best understood as a measure of the...
, b, where 1 b = 10−28 m² = 100 fm².
The scattering of X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
s can also be described in terms of scattering cross-sections, in which case the square ångström
Ångström
The angstrom or ångström, is a unit of length equal to 1/10,000,000,000 of a meter . Its symbol is the Swedish letter Å....
, Ų, is a convenient unit: 1 Ų = 10−20 m² = 104 pm².
Classical Mechanics
In classical mechanicsClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the differential cross section is defined as follows: let a beam of intensity
(measured in number of particles per area per time) be incident on a scattering center. In general, the angle at which a particle is scattered is a function of impact parameterImpact parameter
The impact parameter b is defined as the perpendicular distance between the path of a projectile and the center of the field U created by an object that the projectile is approaching...
. The number of scattered particles per solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
per time,
is therefore well defined. We define the differential cross section to beNote that this quantity has units of area. Furthermore, it depends only on the geometry of the scattering center, and not on the incident flux or distance of the detector from the scattering center. The geometric interpretation is as follows: consider particles that scatter through a solid angle
and ask what values of impact parameter produced them. These impact parameters form a differential area, in space. The differential cross section is simplyQuantum Mechanics
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the wave function of the incident particle is a plane-wave with amplitude 1, that is,
. In general the scattered wave is of the formWe then have as the definition of differential cross section
This has the simple interpretation of the probability of finding a scattered particle within a given solid angle.
The integral cross section is the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the differential cross section on the whole sphere of observation (4
steradianSteradian
The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...
):
A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. Usual units are the cm2, the barn
Barn (unit)
A barn is a unit of area. Originally used in nuclear physics for expressing the cross sectional area of nuclei and nuclear reactions, today it is used in all fields of high energy physics to express the cross sections of any scattering process, and is best understood as a measure of the...
(1 b = 10−28 m2) and the corresponding submultiples: the millibarn (1 mb = 10−3 b), the microbarn (1
b = 10−6 b), the nanobarn ( 1 nb = 10−9 b), the picobarn (1 pb = 10−12 b), and the shed (1 shed = 10−24 b). The cross section of two particlesElementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
(i.e. observed when the two particles are colliding
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
with each other) is a measure of the interaction event between the two particles.
The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles
) depends only on the number of incident particles per unit of time (current of incident particles ), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction. For we haveRelation to the S matrix
If the reduced massReduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
es and momenta
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
of the colliding system are mi,
and mf, before and after the collision respectively, the differential cross section is given bywhere the on-shell T matrix is defined by
in terms of the S matrix
S matrix
In physics, the scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process...
. The
function is the distributionDistribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
called the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
. The computation of the S matrix is the main aim of the scattering theory
Scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
.
See also
- Cross section (physics)Cross section (physics)A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
- Nuclear cross sectionNuclear cross sectionThe nuclear cross section of a nucleus is used to characterize the probability that a nuclear reaction will occur. The concept of a nuclear cross section can be quantified physically in terms of "characteristic area" where a larger area means a larger probability of interaction...
- Neutron cross section
- Scattering amplitudeScattering amplitudeIn quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process...