How to calculate Rutherford Coulomb scattering cross section lets you estimate the probability of elastic deflection of a charged particle in a Coulomb field. The formula was derived in 1911 by Ernest Rutherford (Manchester, Nobel Prize 1908) to explain the Geiger-Marsden gold-foil experiment (1909) — discovering the atomic nucleus. The differential cross section is dσ/dΩ = (Z₁Z₂e²/4E)² · 1/sin⁴(θ/2), where Z₁ and Z₂ are projectile and target atomic numbers, E the kinetic energy in the CM frame, θ the scattering angle. The formula is valid for elastic scattering below the Coulomb barrier in the non-relativistic regime. It underpins RBS (Rutherford Backscattering Spectrometry) at ANL ATLAS, ORNL HRIBF, NIST Gaithersburg, the multiple-Coulomb-scattering modeling in ATLAS, CMS, ALICE, LHCb tracking detectors, and ion-transport calculations in radiotherapy and space dosimetry.

1. The Rutherford formula and Coulomb barrier

The classical formula applies to elastic scattering of point-like charged particles in the non-relativistic regime, below the Coulomb barrier V_C = Z₁Z₂e²/(4πε₀·R), where R is the nuclear radius. For α (Z₁=2) on Au-197 (Z₂=79) at E = 5 MeV, V_C ≈ 26 MeV — below this threshold the projectile sees only the Coulomb field and undergoes no nuclear reactions. The 1/sin⁴(θ/2) characteristic diverges at small angles (forward) and gives the high backscattering probability at 180° observed by Geiger and Marsden. The scattering rate is dN/dt = N_inc · n_t · t · dσ/dΩ · ΔΩ, fundamental for sizing experimental targets at ANL ATLAS, ORNL HRIBF, NSCL Michigan, and FRIB Michigan State.

2. Rutherford cross section for typical cases

• Alpha 5 MeV on gold Au-197 at 90° — Z₁Z₂ = 158, dσ/dΩ ≈ 5.18 barn/sr. The classic Geiger-Marsden 1909 geometry.
• Proton 1 MeV on carbon C-12 at 170° (RBS) — Z₁Z₂ = 6, dσ/dΩ ≈ 0.29 mbarn/sr. Standard at NIST and ANL.
• Alpha 2 MeV on silicon Si-28 at 165° — Z₁Z₂ = 28, dσ/dΩ ≈ 1.5 barn/sr. Typical RBS analysis.
• Proton 100 keV on uranium U-238 at 130° — Z₁Z₂ = 92, dσ/dΩ ≈ 80 barn/sr.
• Carbon C-12 100 MeV on Pb-208 — Z₁Z₂ = 492, well above Coulomb barrier; requires nuclear reactions.
• Pb-208 on Pb-208 at 5 GeV/u (LHC ALICE) — z²=6724, ultraperipheral Coulomb scattering for J/ψ and χ_c photoproduction.

3. Validity range and corrections

The classical formula fails when:

• energy exceeds the Coulomb barrier: E > V_C admits nuclear reactions (capture, fusion, breakup). For α + Au, V_C ≈ 26 MeV; above this dominate inelastic scattering, fusion, fission
• Relativistic effects require the Mott formula for electrons or protons at v ≈ c
• Nuclear spin and magnetic moments introduce spin-dependent corrections (Mott-Schwinger)
• Finite-size effects: for small impact parameters b < R, the projectile penetrates the nucleus, requiring Glauber-Sitenko form factors F(q²). (5) For atomic screening at very low energy, the potential is screened by atomic electrons (Yukawa effective). Geant4, FLUKA, MCNP, and SRIM-TRIM (Ziegler) Monte Carlo codes implement all these corrections

4. Worked numerical examples

Example 1 — Original Geiger-Marsden experiment: α 5 MeV on Au-197, θ = 90°. Z₁Z₂ = 158; sin⁴(45°) = 0.25. dσ/dΩ ≈ 5.18 barn/sr — consistent with original Manchester 1911 measurements that led to the discovery of the nucleus.

Example 2 — RBS thin-film analysis: proton 2 MeV on Au on Si, θ = 165°. NIST tables: dσ/dΩ_Au ≈ 4.9 barn/sr, dσ/dΩ_Si ≈ 0.034 barn/sr. The 144× ratio enables clear elemental discrimination in the RBS spectrum, basis of the technique at NIST, ANL, ORNL, MIT for thin-film and microelectronics analysis.

Example 3 — ALICE Pb-Pb 5 GeV/u, θ = 1°. Z² = 6724; sin⁴(0.5°) ≈ 5.8×10⁻⁹. Coulomb dσ/dΩ is enormous at small angles, dominating ultraperipheral phenomena. UPC γγ → X reactions produce lepton pairs and mesons at hundreds of mbarn/sr at 1° — basis of UPC physics at ALICE, ATLAS, LHCb.

5. Practical applications

• RBS (Rutherford Backscattering Spectrometry): non-destructive analytical technique for thin films and depth profiles. Active at NIST Gaithersburg, ANL ATLAS, ORNL HRIBF, MIT NSE, Sandia. Resolution 1-10 nm, sensitivity ppm
• PIXE (Particle Induced X-ray Emission) for archaeometry — used on Smithsonian, Met Museum NYC artifacts
• Si tracking detectors: ATLAS, CMS, ALICE, LHCb, Belle II use Rutherford for multiple Coulomb scattering modeling
• Hot plasma diagnostics: in tokamaks like ITER (Cadarache) and JT-60SA (Naka, Japan-Europe) elastic neutralizer scattering measures ion density
• Astrophysics: Coulomb capture of solar particles from CME (Coronal Mass Ejections), relevant for ISS astronaut radiation exposure and Artemis mission shield design

6. Comparison with other scattering formulas

The Mott formula (1929) generalizes Rutherford for electrons and relativistic particles, including spin-1/2 and v ≈ c effects. The Mott-Schwinger formula adds nuclear spin and magnetic moments. For extended nuclei, the Glauber-Sitenko formalism includes form factor F(q²) describing nuclear size via Fourier transform of charge density. Coulomb excitation (Alder-Steinwedel 1966) describes nuclear transitions induced by the projectile's Coulomb field. For low-energy electrons (<10 keV), Møller (e⁻e⁻) and Bhabha (e⁺e⁻) require full QED treatment. Codes SRIM-TRIM (Ziegler-Biersack), Geant4, FLUKA, GEANT4-DNA implement all variants.

7. Special cases and extreme regimes

For exotic radioactive nuclei at FRIB Michigan State, RIBF RIKEN, FAIR Darmstadt, GANIL Caen, INFN-LNL SPES, Rutherford is the starting point for measuring elastic reactions of drip-line nuclei. Mott-Coulomb scattering between identical nuclei (¹²C-¹²C, ⁵⁸Ni-⁵⁸Ni) shows quantum interference with dσ/dΩ_Mott = dσ/dΩ_Ruth · (1 + cos(2η ln(tan(θ/2)))). For ultraperipheral collisions (UPC) at LHC Pb-Pb 5.02 TeV, the Z²α² factor makes equivalent-photon flux dominant, basis of J/ψ and η_c photoproduction at ALICE and LHCb. For antiproton scattering p̄ on nuclei (PANDA-FAIR Darmstadt 2027), Rutherford includes z₁=-1 with opposite sign, predicting measurable experimental asymmetries.

Updated July 2026