quantity

cross section

A measure of the probability of a specific nuclear reaction occurring, expressed as an effective area.
19 related articles · 6 formulas

Description

In nuclear and particle physics, the cross section is a measure of the probability that a specific interaction will occur when particles collide. It is expressed as an effective area, typically in units of barns (1 barn = 10^-28 square meters), and represents the 'size' of a target as seen by an incoming projectile. A larger cross section means the reaction is more likely to happen; a smaller one means it is rarer. This concept is central to understanding everything from neutron capture in reactors to the production of heavy particles like the top quark at the Large Hadron Collider.

History

The concept was introduced in the early 20th century, with the term 'cross section' used in scattering experiments by Ernest Rutherford and his colleagues around 1911–1913 to describe the probability of alpha particles being deflected by gold nuclei.

How it works

Imagine firing a beam of particles at a thin foil of target material. The cross section is defined as the ratio of the number of reactions per target nucleus to the number of incoming particles per unit area. It is not a physical size of the nucleus but a quantum-mechanical probability, often influenced by resonances, quantum tunneling, and the energy of the incoming particle. For example, a neutron might have a huge cross section for capture at certain energies due to resonance, making it appear 'large' even though the nucleus itself is tiny.

In practice

Cross sections are used to design nuclear reactors, calculate radiation shielding, and estimate production rates of rare particles in colliders, such as the top quark or Higgs boson.

Fun fact

The unit 'barn' was coined during the Manhattan Project as a slang term for 'big as a barn' to describe the unexpectedly large cross section of certain nuclear reactions.

Concept graph →

Formulas

$$\sigma \propto \frac{\hbar G}{c^3}$$
Cross section scales with Planck area in quantum gravity.
What is in the formula
\sigmavariablecross section — cross section metre squared
\hbarconstantreduced Planck constant, fundamental quantum of action — reduced planck constant = 1.054571817e-34 joule second
GconstantNewton's gravitational constant, determines strength of gravity — gravitational constant = 6.67430e-11 metre cubed per kilogram per second squared
cconstantspeed of light in vacuum, fundamental constant of relativity — speed of light = 299792458 metre per second
\proptooperatorproportional to — proportionality
$$\tau = n \sigma l$$
Optical depth equals column density times cross-section.
What is in the formula
τvariableoptical depth — optical depth
nvariablenumber density of absorbers — number density per cubic metre
σvariablecross-section per absorber — cross section square metre
lvariablepath length through the medium — length metre
$$C \sim \pi R^2 \frac{\rho_\chi}{m_\chi} v \sigma$$
Rate at which stars capture dark matter depends on density and cross-section.
What is in the formula
Cvariablecapture rate of dark matter particles by the star — capture rate per second
Rvariableradius of the star — radius metre
ρ_χvariablemass density of dark matter in the local environment — mass density kilogram per cubic metre
m_χvariablemass of a single dark matter particle — mass kilogram
vvariabletypical relative speed of dark matter particles with respect to the star — velocity metre per second
σvariablescattering cross-section between dark matter and stellar nucleons — cross section square metre
$$I = I_0 e^{-\sigma n x}$$
Intensity decreases exponentially with absorber column density.
What is in the formula
IvariableTransmitted intensity — intensity watt per square metre
I_0variableIncident intensity — intensity watt per square metre
σvariableAbsorption cross-section per absorber — cross section square metre
nvariableNumber density of absorbers — number density per cubic metre
xvariablePath length through the medium — length metre
$$\sigma(E) \propto E^\alpha$$
Cross section scales as power of energy.
What is in the formula
σvariableCross section — cross section metre squared
EvariableEnergy of the incident particle — energy joule
αvariablePower-law exponent — dimensionless exponent
∝operatorProportional to, up to a constant factor — proportionality
$$\sigma = \frac{g_1^2 g_2^2 \mu^2}{\pi m^4}$$
Cross section for contact interaction between particles.
What is in the formula
\sigmavariableCross section — cross section square metre
g_1variableCoupling constant of particle 1 — coupling constant joule metre
g_2variableCoupling constant of particle 2 — coupling constant joule metre
\muvariableReduced mass of the two-particle system — mass kilogram
mvariableMass scale (typically the mass of the particles) — mass kilogram

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