In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle or density fluctuation). For example, the Rutherford cross-section is a measure of probability that an

alpha particle Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be prod ...

will be deflected by a given angle during an interaction with an

atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...

. Cross section is typically denoted (

sigma Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...

) and is expressed in units of area, more specifically in

barn A barn is an agricultural building usually on farms and used for various purposes. In North America, a barn refers to structures that house livestock, including cattle and horses, as well as equipment and fodder, and often grain.Alle ...

s. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

. In

classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...

, this probability often converges to a deterministic proportion of excitation energy involved in the process, so that, for example, with light scattering off of a particle, the cross section specifies the amount of optical power scattered from light of a given irradiance (power per area). It is important to note that although the cross section has the same units as area, the cross section may not necessarily correspond to the actual physical size of the target given by other forms of measurement. It is not uncommon for the actual cross-sectional area of a scattering object to be much larger or smaller than the cross section relative to some physical process. For example, plasmonic nanoparticles can have light scattering cross sections for particular frequencies that are much larger than their actual cross-sectional areas. When two discrete particles interact in classical physics, their mutual cross section is the area

transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle * Transverse flute, a flute that is held horizontally * Transverse force (or ''Euler force''), the tange ...

to their relative motion within which they must meet in order to

scatter Scatter may refer to: * Scattering, in physics, the study of collisions * Statistical dispersion or scatter * Scatter (modeling), a substance used in the building of dioramas and model railways * Scatter, in computer programming, a parameter in ...

from each other. If the particles are hard inelastic

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

s that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-at-a-distance force, such as

electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

, their scattering cross section is generally larger than their geometric size. When a cross section is specified as the differential limit of a function of some final-state variable, such as particle angle or energy, it is called a differential cross section (see detailed discussion below). When a cross section is integrated over all scattering angles (and possibly other variables), it is called a total cross section or integrated total cross section. For example, in Rayleigh scattering, the intensity scattered at the forward and backward angles is greater than the intensity scattered sideways, so the forward differential scattering cross section is greater than the perpendicular differential cross section, and by adding all of the infinitesimal cross sections over the whole range of angles with integral calculus, we can find the total cross section. Scattering cross sections may be defined in

nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space *Nuclear ...

, atomic, and

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

for collisions of accelerated beams of one type of particle with targets (either stationary or moving) of a second type of particle. The probability for any given reaction to occur is in proportion to its cross section. Thus, specifying the cross section for a given reaction is a proxy for stating the probability that a given scattering process will occur. The measured

reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per uni ...

of a given process depends strongly on experimental variables such as the density of the target material, the intensity of the beam, the detection efficiency of the apparatus, or the angle setting of the detection apparatus. However, these quantities can be factored away, allowing measurement of the underlying two-particle collisional cross section. Differential and total scattering cross sections are among the most important measurable quantities in

, atomic, and

Collision among gas particles

In a gas of finite-sized particles there are collisions among particles that depend on their cross-sectional size. The average distance that a particle travels between collisions depends on the density of gas particles. These quantities are related by :

\sigma = \frac,

where : is the cross section of a two-particle collision ( SI units: m²), : is the

mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...

between collisions (SI units: m), : is the

number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...

of the target particles (SI units: m⁻³). If the particles in the gas can be treated as hard spheres of radius that interact by direct contact, as illustrated in Figure 1, then the effective cross section for the collision of a pair is :

\sigma = \pi \left(2r\right)^2

If the particles in the gas interact by a force with a larger range than their physical size, then the cross section is a larger effective area that may depend on a variety of variables such as the energy of the particles. Cross sections can be computed for atomic collisions but also are used in the subatomic realm. For example, in

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies t ...

a "gas" of low-energy

neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...

s collides with nuclei in a reactor or other nuclear device, with a cross section that is energy-dependent and hence also with well-defined

between collisions.

Attenuation of a beam of particles

If a beam of particles enters a thin layer of material of thickness , the

flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...

of the beam will decrease by according to :

\frac = -n \sigma \Phi,

where is the total cross section of ''all'' events, including

scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which s ...

, or transformation to another species. The volumetric number density of scattering centers is designated by . Solving this equation exhibits the exponential attenuation of the beam intensity: :

\Phi = \Phi_0 e^,

where is the initial flux, and is the total thickness of the material. For light, this is called the

Beer–Lambert law The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied t ...

Differential cross section

Consider a classical measurement where a single particle is scattered off a single stationary target particle. Conventionally, a

spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

is used, with the target placed at the origin and the axis of this coordinate system aligned with the incident beam. The angle is the scattering angle, measured between the incident beam and the scattered beam, and the is the

azimuthal angle An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematically, ...

. :

The

impact parameter In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). It is often referred to in ...

is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle . For a given interaction ( Coulombic,

magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particl ...

gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

al, contact, etc.), the impact parameter and the scattering angle have a definite one-to-one functional dependence on each other. Generally the impact parameter can neither be controlled nor measured from event to event and is assumed to take all possible values when averaging over many scattering events. The differential size of the cross section is the area element in the plane of the impact parameter, i.e. . The differential angular range of the scattered particle at angle is the solid angle element . The differential cross section is the quotient of these quantities, . It is a function of the scattering angle (and therefore also the impact parameter), plus other observables such as the momentum of the incoming particle. The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deflection. In cylindrically symmetric situations (about the beam axis), the

is not changed by the scattering process, and the differential cross section can be written as :

\frac =\int_0^ \frac \,\mathrm\varphi

. In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle. For scattering of particles of incident flux off a stationary target consisting of many particles, the differential cross section at an angle is related to the flux of scattered particle detection in particles per unit time by :

\frac(\theta,\varphi) = \frac \frac.

Here is the finite angular size of the detector (SI unit: sr), is the

of the target particles (SI units: m⁻³), and is the thickness of the stationary target (SI units: m). This formula assumes that the target is thin enough that each beam particle will interact with at most one target particle. The total cross section may be recovered by integrating the differential cross section over the full

solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...

( steradians): :

\sigma = \oint_ \frac \, \mathrm d \Omega = \int_0^ \int_0^\pi \frac \sin \theta \, \mathrm d \theta \, \mathrm d \varphi.

It is common to omit the “differential”

qualifier In linguistics, a modifier is an optional element in phrase structure or clause structure which ''modifies'' the meaning of another element in the structure. For instance, the adjective "red" acts as a modifier in the noun phrase "red ball", prov ...

when the type of cross section can be inferred from context. In this case, may be referred to as the ''integral cross section'' or ''total cross section''. The latter term may be confusing in contexts where multiple events are involved, since “total” can also refer to the sum of cross sections over all events. The differential cross section is extremely useful quantity in many fields of physics, as measuring it can reveal a great amount of information about the internal structure of the target particles. For example, the differential cross section of

Rutherford scattering In particle physics, Rutherford scattering is the elastic scattering of charged particles by the Coulomb interaction. It is a physical phenomenon explained by Ernest Rutherford in 1911 that led to the development of the planetary Rutherford model ...

provided strong evidence for the existence of the atomic nucleus. Instead of the solid angle, the momentum transfer may be used as the independent variable of differential cross sections. Differential cross sections in inelastic scattering contain resonance peaks that indicate the creation of metastable states and contain information about their energy and lifetime.

Quantum scattering

In the time-independent formalism of

quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...

scattering, the initial wave function (before scattering) is taken to be a plane wave with definite

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...

: :

\phi_-(\mathbf r) \;\stackrel\; e^,

where and are the ''relative'' coordinates between the projectile and the target. The arrow indicates that this only describes the ''asymptotic behavior'' of the wave function when the projectile and target are too far apart for the interaction to have any effect. After scattering takes place it is expected that the wave function takes on the following asymptotic form: :

\phi_+(\mathbf r) \;\stackrel\; f(\theta,\phi) \frac,

where is some function of the angular coordinates known as the

scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.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 ), and the type of interaction. For we have :

\begin
I_\text &= I_\textN\sigma, \\
\sigma &= \frac \frac \\
&= \text \times \frac.
\end

Relation to the S-matrix

If the reduced masses and

momenta Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles. In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ...

of the colliding system are , and , before and after the collision respectively, the differential cross section is given by :

\frac = \left(2\pi\right)^4 m_i m_f \frac \bigl, T_\bigr, ^2,

where the on-shell matrix is defined by :

S_ = \delta_ - 2\pi i \delta\left(E_f - E_i\right) \delta\left(\mathbf_i - \mathbf_f\right) T_

in terms of the S-matrix. Here is the

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...

. The computation of the S-matrix is the main goal of the

scattering theory In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...

Units

Although the SI unit of total cross sections is m², smaller units are usually used in practice. In nuclear and particle physics, the conventional unit is the barn b, where 1 b = 10⁻²⁸ m² = 100 fm². Smaller prefixed units such as mb and μb are also widely used. Correspondingly, the differential cross section can be measured in units such as mb/sr. When the scattered radiation is visible light, it is conventional to measure the path length in centimetres. To avoid the need for conversion factors, the scattering cross section is expressed in cm², and the number concentration in cm⁻³. The measurement of the scattering of visible light is known as nephelometry, and is effective for particles of 2–50

µm The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer ( American spelling), also commonly known as a micron, is a unit of length in the International System of Uni ...

in diameter: as such, it is widely used in

meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...

and in the measurement of atmospheric pollution. The scattering of

X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...

s can also be described in terms of scattering cross sections, in which case the square

ångström The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...

is a convenient unit: 1 Å² = 10⁻²⁰ m² = = 10⁸ b. The sum of the scattering, photoelectric, and pair-production cross-sections (in barns) is charted as the "atomic attenuation coefficient" (narrow-beam), in barns.

Scattering of light

For light, as in other settings, the scattering cross section for particles is generally different from the geometrical cross section of the particle, and it depends upon the

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...

of light and the permittivity, shape, and size of the particle. The total amount of scattering in a sparse medium is proportional to the product of the scattering cross section and the number of particles present. In the interaction of light with particles, many processes occur, each with their own cross sections, including

, and

photoluminescence Photoluminescence (abbreviated as PL) is light emission from any form of matter after the absorption of photons (electromagnetic radiation). It is one of many forms of luminescence (light emission) and is initiated by photoexcitation (i.e. photo ...

. The sum of the absorption and scattering cross sections is sometimes referred to as the attenuation or extinction cross section. :

\sigma = \sigma_\text + \sigma_\text + \sigma_\text.

The total extinction cross section is related to the attenuation of the light intensity through the

, which says that attenuation is proportional to particle concentration: :

A_\lambda = C l \sigma,

where is the attenuation at a given

, is the particle concentration as a number density, and is the path length. The absorbance of the radiation is the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...

( decadic or, more usually,

natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...

) of the reciprocal of the

transmittance Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is t ...

: :

A_\lambda = -\log \mathcal.

Combining the scattering and absorption cross sections in this manner is often necessitated by the inability to distinguish them experimentally, and much research effort has been put into developing models that allow them to be distinguished, the Kubelka-Munk theory being one of the most important in this area.

Cross section and Mie theory

Cross sections commonly calculated using

Mie theory 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 an electromagnetic plane wave by a homogeneous sphere. The solution takes the ...

include efficiency coefficients for extinction

Q_\text

, scattering

Q_\text

, and Absorption

Q_\text

cross sections. Those are normalized by the geometrical cross sections of the particle

\sigma_\text = \pi a^2

Q_\alpha = \frac, \qquad \alpha = \text, \text, \text.

The cross section is defined by :

\sigma_\alpha = \frac

where

\left_\alpha \right = \left \text \right /math> is the energy flow through the surrounding surface, and \left_\right = \left \frac \right /math> is the intensity of the incident wave. For a

plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...

the intensity is going to be

I_ = , \mathbf, ^2 / (2 \eta)

, where

\eta = \sqrt

is the impedance of the host medium. The main approach is based on the following. Firstly, we construct an imaginary sphere of radius

r

(surface

A

) around the particle (the scatterer). The net rate of electromagnetic energy crosses the surface

A

is :

W_a = - \oint_A \mathbf \cdot \hat dA

where

\mathbf = \frac \operatorname \left \mathbf^* \times \mathbf \right /math> is the time averaged Poynting vector. If W_a > 0 energy is absorbed within the sphere, otherwise energy is being created within the sphere. We exclude from consideration the later one. Once host medium is non-absorbing, the energy is absorbed by the particle. We decompose the total field into incident and scattered parts \mathbf = \mathbf_i + \mathbf_s, and the same for the magnetic field \mathbf . Thus, we can decompose W_a into the three terms W_a = W_i - W_s + W_, where
: W_i = - \oint_A \mathbf_i \cdot \hat dA \equiv 0, \qquad
W_s = \oint_A \mathbf_s \cdot \hat dA, \qquad
W_ = \oint_A \mathbf_ \cdot \hat dA. Where \mathbf_i = \frac \operatorname \left \mathbf_i^* \times \mathbf_i \right, \mathbf_s = \frac \operatorname \left \mathbf_s^* \times \mathbf_s \right, and \mathbf_ = \frac \operatorname \left \mathbf_s^* \times \mathbf_i + \mathbf_i^* \times \mathbf_s \right . 

All the field can be decomposed into the series of vector spherical harmonics (VSH) . After that, all the integrals can be taken.
In the case of a uniform sphere of radius a, permittivity \varepsilon, and permeability \mu the problem has a precise solution.  The scattering and extinction coefficients are Q_\text = \frac\sum_^\infty (2n+1)(, a_, ^2+, b_, ^2) Q_\text = \frac\sum_^\infty (2n+1)\Re(a_+b_) Where k = n_\text k_0 . Those are connected as \sigma_\text = \sigma_\text + \sigma_\text \qquad \text \qquad Q_\text = Q_\text + Q_\text

Dipole approximation for the scattering cross section

Let us assume that particle support only electric and magnetic dipole modes with polarizabilities

\mathbf = \alpha^e \mathbf

and

\mathbf = (\mu \mu_0)^\alpha^m \mathbf

(here we use the notation of magnetic polarizability in the manner of Bekshaev et al. rather than the notation of Nieto-Vesperians et al.) expressed through the Mie coefficients as

\alpha^e = 4 \pi \varepsilon_0 \cdot i \frac a_1, \qquad  \alpha^m = 4 \pi \mu_0 \cdot i \frac b_1.

Then the cross sections are going to be

\sigma_ = \sigma_^ + \sigma_^ =
\frac \cdot 4\pi k \Im(\alpha^e) + 
\frac \cdot 4\pi k \Im(\alpha^m)

\sigma_ = \sigma_^ + \sigma_^ = 
\frac \cdot \frac k^4 , \alpha^e, ^2 +
\frac \cdot \frac k^4 , \alpha^m, ^2

and, finally, the electric and magnetic absorption cross sections

\sigma_ = \sigma_^ + \sigma_^

are

\sigma_^ = \frac \cdot 4\pi k \left \alpha^e, ^2\right

and

\sigma_^ = \frac \cdot 4\pi k \left \alpha^m, ^2\right

For the case of no-inside-gain particle, i.e. there no energy is emitted by the particle internally (

\sigma_ > 0

), we have a particular case of the Optical theorem

\frac \Im(\alpha^e) + \frac \Im(\alpha^m) \geq \frac \left \frac  + \frac \right

The sign of equality

\geq \to =

is achieved for non-absorbing particles, i.e. for

\Im(\varepsilon) = \Im(\mu) = 0

Scattering of light on extended bodies

In the context of scattering light on extended bodies, the scattering cross section, , describes the likelihood of light being scattered by a macroscopic particle. In general, the scattering cross section is different from the geometrical cross section of a particle, as it depends upon the wavelength of light and the permittivity in addition to the 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'' () is the sum of the cross sections due to

, scattering, and

luminescence Luminescence is spontaneous emission of light by a substance not resulting from heat; or "cold light". It is thus a form of cold-body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions or stress on a crys ...

: :

\sigma = \sigma_\text + \sigma_\text + \sigma_\text.

The total cross section is related to the

absorbance Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...

of the light intensity through the

, which says that absorbance is proportional to concentration: , where is the absorbance at a given

, is the concentration as a

, and is the path length. The extinction or

of the radiation is the

( decadic or, more usually,

) of the reciprocal of the

: :

A_\lambda = - \log \mathcal.

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 looking at a halo surrounding the moon on a decently foggy evening: Red light photons experience a larger cross sectional area of water droplets than photons of higher energy do. The halo around the moon thus has a perimeter of red light due to lower energy photons being scattering further from the center of the moon. Photons from the rest of the visible spectrum are left within the center of the halo and perceived as white light.

Meteorological range

The scattering cross section is related to the meteorological range : :

L_\text = \frac.

The quantity is sometimes denoted , the scattering coefficient per unit length.

Examples

Example 1: elastic collision of two hard spheres

The

elastic collision In physics, an elastic collision is an encounter ( collision) between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into ...

of two hard spheres is an instructive example that demonstrates the sense of calling this quantity a cross section. and are respectively the radii of the scattering center and scattered sphere. The total cross section is :

\sigma_\text = \pi \left(r + R\right)^2.

So in this case the total scattering cross section is equal to the area of the circle (with radius ) within which the center of mass of the incoming sphere has to arrive for it to be deflected, and outside which it passes by the stationary scattering center. When the radius of the incoming sphere is approaching zero, the cross section is just the area of a circle with radius R.

Example 2: scattering light from a 2D circular mirror

Another example illustrates the details of the calculation of a simple

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...

scattering model obtained by a reduction of the dimension. For simplicity, we will consider the scattering of a beam of light on a plane treated as a uniform density of parallel rays and within the framework of geometrical optics from a circle with radius with a perfectly reflecting boundary. Its three-dimensional equivalent is therefore the more difficult problem of a laser or flashlight light scattering from the mirror sphere, for example, from the mechanical bearing ball. The unit of cross section in one dimension is the unit of length, for example 1 m. Let be the angle between the light ray and the

radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

joining the reflection point of the light ray with the center point of the circle mirror. Then the increase of the length element perpendicular to the light beam is expressed by this angle as :

\mathrm dx = r \cos \alpha \,\mathrm d \alpha,

the reflection angle of this ray with respect to the incoming ray is then , and the scattering angle is :

\theta = \pi - 2 \alpha.

The energy or the number of photons reflected from the light beam with the intensity or density of photons on the length is :

I \,\mathrm d \sigma = I \,\mathrm dx(x) = I r \cos \alpha \,\mathrm d \alpha = I \frac \sin \left(\frac\right) \,\mathrm d \theta = I \frac \,\mathrm d \theta.

The differential cross section is therefore () :

\frac = \frac \sin \left(\frac\right).

As it is seen from the behaviour of the

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...

function, this quantity has the maximum for the backward scattering (; the light is reflected perpendicularly and returns), and the zero minimum for the scattering from the edge of the circle directly forward (). It confirms the intuitive expectations that the mirror circle acts like a diverging lens, and a thin beam is more diluted the closer it is from the edge defined with respect to the incoming direction. The total cross section can be obtained by summing (integrating) the differential section of the entire range of angles: :

\sigma = \int_0^ \frac \,\mathrm d \theta = \int_0^ \frac \sin \left(\frac\right) \,\mathrm d \theta = \left. -r \cos \left(\frac\right) \_0^ = 2 r,

so it is equal as much as the circular mirror is totally screening the two-dimensional space for the beam of light. In three dimensions for the mirror ball with the radius it is therefore equal .

Example 3: scattering light from a 3D spherical mirror

We can now use the result from the Example 2 to calculate the differential cross section for the light scattering from the perfectly reflecting sphere in three dimensions. Let us denote now the radius of the sphere as . Let us parameterize the plane perpendicular to the incoming light beam by the cylindrical coordinates and . In any plane of the incoming and the reflected ray we can write now from the previous example: :

\begin
r &= a \sin \alpha,\\
\mathrm dr &= a \cos \alpha \,\mathrm d \alpha,
\end

while the impact area element is :

\mathrm d \sigma = \mathrm d r(r) \times r \,\mathrm d \varphi = \frac \sin \left(\frac\right) \cos \left(\frac\right) \,\mathrm d \theta \,\mathrm d \varphi.

Using the relation for the solid angle in the spherical coordinates: :

\mathrm d\Omega = \sin \theta \,\mathrm d \theta \,\mathrm d \varphi

and the trigonometric identity :

\sin \theta = 2 \sin \left(\frac\right) \cos \left(\frac\right),

we obtain :

\frac = \frac,

while the total cross section as we expected is :

\sigma = \oint_ \frac \,\mathrm d \Omega = \pi a^2.

As one can see, it also agrees with the result from the Example 1 if the photon is assumed to be a rigid sphere of zero radius.