How to calculate Mott elastic electron-nucleus scattering

Compute Mott differential cross-section dσ/dΩ for elastic electron-nucleus scattering (Sir Nevill Mott Cambridge 1929). Z=79 gold, recoil, cos²(θ/2) factor — JLab Hall A, INFN-Rome, DAFNE Frascati.

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The Mott scattering calculator determines the differential cross-section of relativistic elastic scattering of an electron from a nucleus, via the Mott formula: dσ/dΩ = (Z² α² (ℏc)² / 4 p² β²) · cos²(θ/2) / sin⁴(θ/2). Developed by Sir Nevill Francis Mott (Cambridge University 1929 + Bristol 1933) in "The scattering of fast electrons by atomic nuclei" Proc. Roy. Soc. A 124:425, the formula generalizes classical Rutherford scattering 1911 including relativistic and spin 1/2 effects. The cos²(θ/2) factor distinguishes Mott from Rutherford: for backward scattering (θ=180°) the cross-section is zero — Dirac electrons cannot completely change direction conserving spin. Mott won the 1977 Nobel Prize with Anderson and Van Vleck for work on non-crystalline solids (Mott insulator), but this 1929 formula remains foundational in nuclear physics.

1. Mott 1929 formula

The complete differential cross-section is dσ/dΩ = (Z² α² (ℏc)² / 4 p² β²) · cos²(θ/2) / sin⁴(θ/2), with Z target nucleus atomic number, α ≈ 1/137 Sommerfeld fine structure constant, p electron momentum in MeV/c, β = v/c relativistic velocity, θ scattering angle. The non-relativistic limit (β << 1) recovers Rutherford 1911 dσ/dΩ = (Z e²/(4 E sin²(θ/2)))² · cos²(θ/2). The cos²(θ/2) factor derives from Dirac spin conservation: at θ=180° (back-scattering) it vanishes. The ultra-relativistic limit (γ >> 1, β → 1): dσ/dΩ ∝ cos²(θ/2)/[E² sin⁴(θ/2)]. For extended nuclei add the form factor F(q²): dσ/dΩ_complete = dσ/dΩ_Mott · |F(q²)|².

2. Typical parameters and common targets

  • Hydrogen H: Z=1, primary target, proton-sensitive, JLab Hall A measures G_E^p, G_M^p form factors.
  • Carbon C: Z=6, light target, quasi-elastic scattering, JLab.
  • Oxygen O: Z=8, doubly magic Z=8, nuclear physics target.
  • Aluminum Al: Z=13, mid-light target for spectroscopy.
  • Iron Fe: Z=26, doubly magic N=Z=28, neutron form factor.
  • Gold Au: Z=79, classical Rutherford-Mott target, high cross-section.
  • Lead Pb: Z=82, doubly magic Z=82 N=126, ²⁰⁸Pb stable isotope.
  • Uranium U: Z=92, heaviest stable target, possible fission at high energy.

3. Range of validity of Mott

The Mott formula holds for pure elastic scattering (kinetic energy conserved, no inelastic excitation) of electrons from point-mass nuclei. Limits: (a) Extended nucleus: at momentum transfer q > 100 MeV/c, λ_dB becomes < nuclear radius R_N ≈ 1.2 A^(1/3) fm. Introduce form factor F(q²): dσ/dΩ_real = dσ/dΩ_Mott · |F(q²)|². (b) Inelastic excitation: above threshold (5-10 MeV), electrons excite nuclear states, inelastic scattering. (c) Bremsstrahlung: at high energy, radiation energy loss. (d) Coulomb distortion: for high Z (>50), Born approximation corrections fail. Use partial wave analysis. (e) Deep Inelastic Scattering DIS: at Q² > 1 GeV², electron resolves quarks, parton regime (Bjorken scaling), Mott no longer holds.

4. Worked numerical examples

Example 1 — 100 MeV electron on Au (Z=79) at θ=60°. p = √(100.5² - 0.511²) = 100.5 MeV/c. β = 0.9999. cos(30°) = 0.866, sin(30°) = 0.5. dσ/dΩ_Mott = (79²·(1/137)²·197.3² / 4·100.5²·0.9999²)·(0.866²/0.5⁴) = 0.107 · 12 = 1.28 mb/sr. Classical Geiger-Marsden 1909 target (Rutherford α-Au), Mott electron 1929.

Example 2 — 1 GeV electron on Pb (Z=82) at θ=30°. p ≈ 1000 MeV/c. cos(15°) = 0.966, sin(15°) = 0.259. dσ/dΩ_Mott ≈ 4.1 mb/sr. JLab Hall A measured dσ/dΩ for ²⁰⁸Pb at 1 GeV, yielding form factor F(q²) and nuclear density ρ(r).

Example 3 — 10 MeV electron on H (Z=1) at θ=90°. p ≈ 10.5 MeV/c. β = 0.9988. dσ/dΩ_Mott ≈ 0.000018 mb/sr = 18 nb/sr. Tiny cross-section (Z=1), JLab experiments measure at high luminosity.

5. Italian experiments using Mott scattering

  • JLab Hall A INFN-Rome collaboration: 5-12 GeV electrons on H, He-3, He-4, Pb nuclei. Measures G_E^p, G_M^p, G_E^n, G_M^n electromagnetic form factors of proton and neutron. Sapienza Rome has strong groups since 1994.
  • INFN-LNF Frascati DAFNE 1999: e+e- 510 MeV for phi-mesons, also secondary electron beam fixed-target scattering.
  • Mainz MAMI Germany (Italian participation): 855 MeV cw electron beam, A1/A4 Compton + parity-violating scattering experiments.
  • BlueCRAB ALS Berkeley + ELI-NP Magurele Romania (INFN partner): laser-driven GeV-class electron beam.
  • Pancini-Conversi-Piccioni Rome 1947: first electron-muon (mesotron) cross-section, Yukawa+Mott framework.
  • JLab CEBAF 12 GeV upgrade 2014: parity-violating electrons on Pb (PREX-2 2019), neutron skin thickness mapping, INFN participation.
  • EIC Brookhaven (Electron-Ion Collider) 2030: 100 GeV electrons on nuclei, INFN-Genoa + INFN-Rome participation.
  • Cineca Bologna 1969 Marconi100 + Leonardo HPC 250 PFlops + INFN-CNAF: Monte Carlo Geant4 + GEMC + EIC simulations with Mott scattering kernel.

6. Comparison with other scattering formulas

Mott 1929: electron on nucleus, relativistic elastic scattering, cos²(θ/2) factor. Rutherford 1911: alpha on nucleus, classical non-relativistic, σ ∝ 1/sin⁴(θ/2). Klein-Nishina 1929: photon on electron (Compton scattering), relativistic QED. Compton 1923: non-relativistic Klein-Nishina limit, σ_T = 6.65×10⁻²⁵ cm². Bhabha 1936: e+e- (Bhabha scattering) used for LEP/LEP2 calibration, includes annihilation channel. Møller 1932: e-e- elastic, parity-violation test SLAC E158. DIS Bjorken-Feynman 1969: deep inelastic, parton model, Bjorken scaling. Hofstadter form factor 1956: electron on extended nuclei, Nobel 1961. QED Schwinger-Tomonaga-Feynman 1948: α^n corrections to Mott scattering, Lamb shift and g-2. Mott is the workhorse for classical high-energy electron-nucleus scattering.

7. Modern developments and corrections

Modern corrections to Mott scattering include: Hofstadter form factors (Stanford 1956 Nobel 1961) for extended nuclei, F(q²) measured up to q²=20 GeV² JLab/SLAC. Sachs G_E G_M (Ernst-Sachs-Wali 1960): separated electric/magnetic electromagnetic forms. Two-photon exchange (TPE) Guichon-Vanderhaeghen 2003: corrects Rosenbluth-polarization JLab discrepancy. Higher-order QED: α^n vertex corrections, Lamb shift effects on Coulomb nucleus. Generalized Parton Distributions GPDs Müller-Robaschik-Geyer 1994: 3D nucleon imaging via DVCS (deeply virtual Compton scattering). Lattice QCD: form factors from first principles, RBC/UKQCD + INFN Pisa Cnaf. Software: SIMC JLab, GEMC JLab, Geant4 CERN (M. Asai), FLUKA CERN-INFN, MadGraph + Pythia for Monte Carlo. HPC: Cineca Leonardo 250 PFlops + INFN-CNAF + JLab Lustre. Future experiments: EIC Brookhaven 2030, JLab 22 GeV upgrade, MOLLER SLAC precision parity test.

Updated July 2026

Frequently Asked Questions

What is Mott scattering?

Mott scattering is the relativistic elastic scattering of an electron from a nucleus, described by dσ/dΩ = (Z² α² (ℏc)² / 4 p² β²) · cos²(θ/2)/sin⁴(θ/2). Developed by Sir Nevill Mott (Cambridge 1929 + Bristol 1933) as relativistic generalization of Rutherford 1911. The cos²(θ/2) factor derives from Dirac spin conservation. Mott Nobel 1977 (for Mott insulator).

What is the difference between Mott and Rutherford?

Rutherford 1911 holds for alpha particles (no spin) non-relativistic, σ ∝ 1/(sin⁴(θ/2)·E²). Mott 1929 extends to relativistic electrons with spin 1/2: adds cos²(θ/2) factor that vanishes at θ=180° (back-scattering). For E < 1 MeV β < 100 MeV relativistic effects dominate.

How is Mott total cross-section computed?

Total cross-section has singularity at θ=0 (forward): ∫dσ/dΩ dΩ ∝ ∫(cos²(θ/2)/sin⁴(θ/2)) sin θ dθ diverges logarithmically. Introduce cutoff θ_min > 0 (physically: atomic Coulomb screening for |b| > a_Bohr). For Au at 100 MeV with θ_min=1°: σ_tot ≈ 100 b. For real applications (target thickness): effective σ.

What are nuclear form factors?

For extended nuclei (R_N ≈ 1.2 A^(1/3) fm) Mott formula must be multiplied by |F(q²)|², where F(q²) is the Sachs G_E G_M form factor (Hofstadter 1956 Nobel 1961). For proton: G_E^p ≈ G_M^p/μ_p ≈ (1+q²/0.71 GeV²)⁻². Measured JLab Hall A 5-12 GeV, BaBar 13 GeV, EIC Brookhaven 2030 will reach 100 GeV electron-ion.

What is the high-energy limit of Mott?

At E_e > 5 GeV electrons resolve quarks inside proton (deep inelastic scattering DIS), no longer elastic. Bjorken-Feynman 1969 introduced x_B (parton momentum fraction) and Q² (4-momentum transfer). Bjorken scaling F2(x,Q²) → F2(x) at high Q² confirms quark existence. Nobel 1990 Friedman-Kendall-Taylor SLAC.

Which Italian experiments use Mott scattering?

JLab Hall A INFN-Rome collaboration (G_E^p, G_M^p proton form factor), Mainz MAMI (parity-violating PREX-2), Cineca Bologna 1969 Leonardo HPC 250 PFlops Monte Carlo Geant4/GEMC, Pancini-Conversi-Piccioni Rome 1947 historical, EIC Brookhaven 2030 with INFN-Genoa + INFN-Rome + INFN-Frascati participation (electron-ion collider).

What was Mott's 1977 Nobel?

Sir Nevill Mott won 1977 Nobel with Philip Anderson and John Van Vleck for 'electronic structure of magnetic and disordered systems' (Mott insulator, spin glasses, amorphous semiconductors), NOT for the 1929 formula (though it remains foundational). Mott was at Bristol in the 1930s, then Cambridge from 1954. His 1929 formula is one of modern nuclear physics' pillars.

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