The Scattering Cross Section

Definition

The cross section is defined operationally, by counting reactions. Fire a uniform beam at a thin target and measure how often a given process happens:

σ = (reactions per target per second) / (incident flux)

The incident flux Φ is the number of beam particles crossing unit area per unit time, in m⁻²s⁻¹. Dividing a rate (s⁻¹) by a flux (m⁻²s⁻¹) leaves an area (m²) — so the cross section literally is an area, even though nothing about the definition demands it match the particles' physical size.

For a thin target of number density n (targets per m³) and thickness x, a beam of N₀ particles produces

N_scattered = N₀ · n · x · σ

which is how σ is extracted in a real experiment. Thick targets attenuate exponentially, N = N₀·exp(−n σ x), and the quantity Λ = 1/(nσ) is the mean free path between interactions.

Barns, and why the unit is enormous-sounding but tiny

The natural unit is the barn:

1 barn (b) = 10⁻²⁸ m² = 10⁻²⁴ cm² = 100 fm²

That is roughly the cross-sectional area of a heavy nucleus like uranium. The name is a wartime joke: Manhattan Project physicists studying neutron capture found that, to a fast neutron, a nucleus of this size was "as big as a barn." The full sub-unit ladder is in constant use across nuclear and particle physics.

Unit	Value	Typical use
barn (b)	10⁻²⁸ m²	Nuclear reactions, neutron capture
millibarn (mb)	10⁻³¹ m²	Total pp scattering ≈ 100 mb at LHC
microbarn (µb)	10⁻³⁴ m²	Hadronic resonance production
nanobarn (nb)	10⁻³⁷ m²	W and Z boson production
picobarn (pb)	10⁻⁴⁰ m²	Higgs (≈ 49 pb gg→H at 13 TeV), top pairs
femtobarn (fb)	10⁻⁴³ m²	Rare Higgs decays; integrated L in fb⁻¹
attobarn (ab)	10⁻⁴⁶ m²	New-physics search reach; rarest measured processes

Differential and total cross section

Detectors do not usually catch everything — they sit at particular angles. So the richer quantity is the differential cross section, the fraction of scattering that goes into a small cone of solid angle dΩ around a direction (θ, φ):

dσ/dΩ   [units: barns per steradian, b/sr]

Here θ is the polar scattering angle from the beam axis, φ the azimuth, and dΩ = sin θ dθ dφ the element of solid angle. Integrating over the whole sphere (4π sr) collapses the angular detail into a single number, the total cross section:

σ_tot = ∫ (dσ/dΩ) dΩ = ∫₀^{2π} ∫₀^{π} (dσ/dΩ) sin θ dθ dφ

The differential form encodes the physics — the shape of the potential, quantum interference between paths, spin correlations — while σ_tot fixes the overall event rate. Rutherford's 1911 analysis of alpha particles on gold is the archetype: its differential cross section

dσ/dΩ = ( Z₁ Z₂ e² / (16 π ε₀ E) )² · 1 / sin⁴(θ/2)

diverges as θ → 0 (the Coulomb force has infinite range, so the total Coulomb cross section is formally infinite), and its sharp rise at large angles is what revealed the tiny, dense atomic nucleus. Symbols: Z₁, Z₂ are the charges in units of e; e = 1.602 × 10⁻¹⁹ C; ε₀ = 8.854 × 10⁻¹² F/m; E is the kinetic energy of the incident particle; θ is the scattering angle.

The link to the scattering amplitude

Quantum mechanically, scattering is a wave problem. Far from the target, the wavefunction is an incoming plane wave plus an outgoing spherical wave:

ψ(r, θ) ≈ e^{ikz} + f(θ) · e^{ikr} / r

where k = p/ℏ = 2π/λ is the wavenumber and f(θ) is the scattering amplitude, which has units of length. The differential cross section is simply its modulus squared:

dσ/dΩ = |f(θ)|²

So a cross section is a quantum probability wearing the disguise of an area — |f(θ)|² is the probability density for scattering into direction θ. Two more results tie the picture together. Conservation of probability (unitarity) forces the optical theorem,

σ_tot = (4π / k) · Im f(0)

relating the total cross section to the imaginary part of the forward (θ = 0) amplitude — the beam must lose exactly the flux it scatters away. And the amplitude decomposes into partial waves of definite angular momentum ℓ, each contributing at most

σ_ℓ ≤ (2ℓ + 1) · 4π / k²    (the unitarity limit)

which is why a long-wavelength (small-k) particle can carry an enormous cross section: the 4π/k² prefactor blows up as k → 0.

Luminosity × cross section = event rate

At a collider there is no fixed target — two beams pass through each other. The relevant quantity is the luminosity L, and the master equation of collider physics is beautifully simple:

dN/dt = L · σ

L has units of cm⁻²s⁻¹ (it counts how densely the beams overlap), σ is the cross section, and dN/dt is the rate of events of that process. Accumulating over a run, the total number of events is

N = σ · ∫ L dt = σ · (integrated luminosity)

Integrated luminosity ∫L dt is quoted in inverse barns — most often fb⁻¹. This is the single most useful sentence in the whole subject: cross section is how rare the physics is; luminosity is how you gather enough of it.

Quantity	Symbol / value	Meaning
LHC peak luminosity	≈ 2 × 10³⁴ cm⁻²s⁻¹	Beam overlap density
Run 2 integrated L	≈ 140 fb⁻¹ per experiment	Data collected 2015–2018
Total pp cross section (13 TeV)	≈ 110 mb	All inelastic + elastic scattering
Higgs production (gg→H, 13 TeV)	≈ 49 pb	Dominant Higgs channel (all channels ≈ 56 pb)
Higgs events in 140 fb⁻¹	N = 49 pb × 140 fb⁻¹ ≈ 6.9 M	Before branching ratios/efficiency
tt̄ pair production (13 TeV)	≈ 830 pb	Top-quark studies

Resonances and the Breit–Wigner peak

Plot a cross section against collision energy and you often see sharp spikes. Each is a resonance: at that energy the colliding particles briefly form a short-lived state — a compound nucleus, an excited hadron, or an unstable particle — which then decays. Near a resonance of energy E₀ and full width Γ, the cross section follows the (relativistic) Breit–Wigner line shape:

σ(E) ∝ 1 / [ (E − E₀)² + (Γ/2)² ]

The peak sits at E₀; the full width at half maximum is Γ. By the energy–time uncertainty relation, the width and the state's lifetime are reciprocal:

Γ = ℏ / τ    (ℏ = 6.582 × 10⁻¹⁶ eV·s)

A narrow peak means a long-lived state; a broad one means it decays almost instantly. Two famous examples: the J/ψ meson (discovered simultaneously by Ting at Brookhaven and Richter at SLAC in November 1974) appears as an extraordinarily narrow spike in e⁺e⁻ cross sections at 3.097 GeV with Γ ≈ 93 keV — so narrow it implied a new, long-lived charm quark. The Z boson resonance at 91.19 GeV has Γ ≈ 2.50 GeV; scanning its exact width at LEP counted the number of light neutrino species as 2.984 ± 0.008 — that is, exactly three.

Worked example: neutron mean free path in water

Thermal neutrons in water are captured mainly by hydrogen, with a capture cross section σ ≈ 0.33 barn = 0.33 × 10⁻²⁸ m². How far does a neutron travel before capture?

Water has ≈ 3.34 × 10²⁸ molecules per m³, hence n_H ≈ 6.7 × 10²⁸ hydrogen nuclei per m³. The mean free path is

Λ = 1 / (n σ) = 1 / (6.7×10²⁸ · 0.33×10⁻²⁸) ≈ 0.45 m

So a thermal neutron wanders roughly half a metre through water before an average hydrogen nucleus swallows it — which is exactly why water and paraffin make good neutron shields and moderators. Swap in a different isotope and the number changes wildly: for ¹³⁵Xe the capture cross section is about 2.6 × 10⁶ barn, a resonance so large it can poison a nuclear reactor within hours of shutdown.

JavaScript — cross section calculations

const BARN = 1e-28;         // m^2 per barn
const HBAR_eV = 6.582e-16;  // eV*s

// Extract cross section from a thin-target experiment
function crossSection(scattered, incident, n, thickness) {
  // N_scattered = N0 * n * x * sigma  ->  sigma = N_s / (N0 * n * x)
  return scattered / (incident * n * thickness); // m^2
}

// Mean free path between interactions
function meanFreePath(n, sigma_barn) {
  return 1 / (n * sigma_barn * BARN); // metres
}

// Thermal neutron in water (hydrogen capture ~0.33 b)
const n_H = 6.7e28; // hydrogen nuclei per m^3
console.log(`Neutron MFP in water: ${meanFreePath(n_H, 0.33).toFixed(2)} m`); // ~0.45 m

// Event yield from luminosity x cross section
function eventCount(sigma_pb, integratedL_invFb) {
  // 1 fb^-1 * 1 pb = 1000 events  (since 1 pb = 1000 fb)
  return sigma_pb * 1000 * integratedL_invFb;
}

// Higgs (gg->H) ~49 pb over 140 fb^-1 of LHC Run 2 data
console.log(`Higgs events: ${eventCount(49, 140).toLocaleString()}`); // ~6,860,000

// Rutherford differential cross section (SI)
function rutherfordDsigmaDomega(Z1, Z2, E_joule, thetaRad) {
  const e = 1.602e-19, eps0 = 8.854e-12;
  const pref = (Z1 * Z2 * e * e) / (16 * Math.PI * eps0 * E_joule);
  return pref * pref / Math.pow(Math.sin(thetaRad / 2), 4); // m^2/sr
}

// Breit-Wigner: state lifetime from resonance width
function lifetimeFromWidth(gamma_eV) {
  return HBAR_eV / gamma_eV; // seconds
}

// J/psi width ~93 keV
console.log(`J/psi lifetime: ${lifetimeFromWidth(93e3).toExponential(2)} s`); // ~7e-21 s

// Unitarity limit for partial wave l at wavenumber k (1/m)
function unitarityLimit(l, k) {
  return (2 * l + 1) * 4 * Math.PI / (k * k); // m^2
}

Where cross sections show up

• Collider discovery physics. Every new particle — the W, Z, top, Higgs — was announced as a bump or excess in a measured cross section. σ × ∫L dt sets how many candidate events you have.
• Nuclear reactors. Fission, capture, and scattering cross sections (heavily energy-dependent) determine criticality, control-rod worth, and shielding thickness.
• Neutrino astronomy. Neutrino cross sections near 10⁻³⁸ cm² per GeV are why detectors like IceCube need a cubic kilometre of ice to catch a handful of events.
• Astrophysics. Thomson scattering (σ_T = 6.65 × 10⁻²⁹ m²) sets the opacity of the early universe and fixes the last-scattering surface of the CMB.
• Dark-matter searches. Direct-detection experiments quote limits as WIMP–nucleon cross sections, now probing below 10⁻⁴⁷ cm².
• Medical and materials physics. Photon attenuation (photoelectric, Compton, pair-production cross sections) governs X-ray imaging and radiation dosimetry.

Common mistakes

• Assuming σ equals the physical size. Cross section is a wave probability, not a silhouette. Near a resonance it can exceed the geometric area by a factor of a million; for neutrinos it is astronomically smaller.
• Confusing dσ/dΩ with σ_tot. One is per steradian (an angular density), the other is the integral over all 4π. You must multiply by solid angle and integrate to convert.
• Mixing up luminosity and integrated luminosity. Instantaneous L (cm⁻²s⁻¹) gives a rate; integrated ∫L dt (fb⁻¹) gives a total event count. dN/dt = Lσ versus N = σ∫L dt.
• Forgetting the Jacobian. When converting dσ/dΩ from the centre-of-mass frame to the lab frame, the solid-angle transformation introduces a nontrivial factor — angles do not map trivially between frames.
• Reading Γ as a lifetime. Γ is an energy (a width), not a time. The lifetime is τ = ℏ/Γ. A 2.5 GeV width (the Z) means τ ≈ 3 × 10⁻²⁵ s.
• Treating the total Coulomb cross section as finite. The 1/sin⁴(θ/2) Rutherford law diverges at θ = 0 because the Coulomb force is infinite-ranged; only screened or short-range potentials give a finite σ_tot.