Cross section (physics)
Encyclopedia
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
.
In nuclear
and particle physics
, the concept of a cross section is used to express the likelihood of interaction between particles.
When particles in a beam are thrown against a foil made of a certain substance, the cross section
is a hypothetical area
measure around the target particles of the substance (usually its atoms) that represents a surface. If a particle of the beam crosses this surface, there will be some kind of interaction.
The term is derived from the purely classical
picture of (a large number of) point-like
projectiles directed to an area that includes a solid target. Assuming that an interaction will occur (with 100% probability) if the projectile hits the solid, and not at all (0% probability) if it misses, the total interaction probability for the single projectile will be the ratio of the area of the section of the solid (the cross section, represented by
) to the total targeted area.
This basic concept is then extended to the cases where the interaction probability in the targeted area assumes intermediate values - because the target itself is not homogeneous, or because the interaction is mediated by a non-uniform field. A particular case is scattering
.
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
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
:
, it is convenient to express the probability of a particular event by a cross section. Statistically, the centers of the atoms in a thin foil can be considered as points evenly distributed over a plane. The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance
of one of these points. In fact, if there are 
atomic centers in an area 
of the plane, this probability is 
, which is simply the ratio of the aggregate area of circles of radius 
drawn around the points to the whole area. If we think of the atoms as impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this ratio is the probability that the bullet will strike a steel disc, i.e., that the atomic projectile will be stopped by the foil. If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms. This notion can be extended to any interaction between the impinging particle and the atoms in the target. For example, the probability that an alpha particle
striking a beryllium
target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.
Beer-Lambert law
In optics, the Beer–Lambert law, also known as Beer's law or the Lambert–Beer law or the Beer–Lambert–Bouguer law relates the absorption of light to the properties of the material through which the light is travelling.-Equations:The law states that there is a logarithmic dependence between the...
.
In nuclear
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
and particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, the concept of a cross section is used to express the likelihood of interaction between particles.
When particles in a beam are thrown against a foil made of a certain substance, the cross section
is a hypothetical 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...
measure around the target particles of the substance (usually its atoms) that represents a surface. If a particle of the beam crosses this surface, there will be some kind of interaction.
The term is derived from the purely classical
Classical 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...
picture of (a large number of) point-like
Point particle
A point particle is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space...
projectiles directed to an area that includes a solid target. Assuming that an interaction will occur (with 100% probability) if the projectile hits the solid, and not at all (0% probability) if it misses, the total interaction probability for the single projectile will be the ratio of the area of the section of the solid (the cross section, represented by
) to the total targeted area.This basic concept is then extended to the cases where the interaction probability in the targeted area assumes intermediate values - because the target itself is not homogeneous, or because the interaction is mediated by a non-uniform field. A particular case is 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...
.
Scattering
The scattering cross-section, σscat, is a hypothetical area which describes the likelihood of light (or other radiation) being scatteredScattering
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
is the path lengthPath 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...
:
Nuclear physics
In nuclear physicsNuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
, it is convenient to express the probability of a particular event by a cross section. Statistically, the centers of the atoms in a thin foil can be considered as points evenly distributed over a plane. The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance
of one of these points. In fact, if there are atomic centers in an area of the plane, this probability is , which is simply the ratio of the aggregate area of circles of radius drawn around the points to the whole area. If we think of the atoms as impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this ratio is the probability that the bullet will strike a steel disc, i.e., that the atomic projectile will be stopped by the foil. If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms. This notion can be extended to any interaction between the impinging particle and the atoms in the target. For example, the probability that an alpha particleAlpha particle
Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus, which is classically produced in the process of alpha decay, but may be produced also in other ways and given the same name...
striking a beryllium
Beryllium
Beryllium is the chemical element with the symbol Be and atomic number 4. It is a divalent element which occurs naturally only in combination with other elements in minerals. Notable gemstones which contain beryllium include beryl and chrysoberyl...
target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.
See also
- Scattering theoryScattering theoryIn 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...
- RadarRadarRadar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
: The (monostatic) radar cross sectionRadar cross sectionRadar cross section is a measure of how detectable an object is with a radar. A larger RCS indicates that an object is more easily detected.An object reflects a limited amount of radar energy...
is defined as 4 π times the radioRadioRadio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...
differential cross section at 180 degrees. - Cross sectional area
- Rutherford scatteringRutherford scatteringIn physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1911, and led to the development of the Rutherford model of the atom, and eventually to the Bohr model. It is now exploited by the materials analytical technique Rutherford backscattering...
- neutron cross section