Rutherford Scattering Calculations: Cross-Sections and Angular Distributions### Introduction
Rutherford scattering—the deflection of charged particles by the Coulomb field of an atomic nucleus—was pivotal in establishing the nuclear model of the atom. Beyond its historical importance, Rutherford scattering remains a cornerstone in classical and quantum scattering theory. This article develops the essential calculations used to describe scattering by a Coulomb potential, derives the differential and total cross-sections, examines angular distributions, and discusses limitations and modern contexts where Rutherford-like formulas are applied.
Physical setup and assumptions
We consider a beam of charged particles (commonly alpha particles of charge Ze_p where Z is projectile charge number, though often treated simply as +2e) incident on stationary atomic nuclei of charge Ze (where Z is the target nucleus charge number). Key assumptions in the classical Rutherford derivation:
- The interaction is purely Coulombic: V® = k Qq / r = (1/(4πϵ0)) (Z_t e)(Z_p e) / r.
- Target nuclei are much heavier than the projectile (so targets are effectively fixed).
- The projectile motion can be treated classically (valid when de Broglie wavelength ≪ impact parameter).
- No screening by atomic electrons (valid for high-energy projectiles where close approach probes nucleus; otherwise include screening corrections).
Notation (SI unless otherwise noted):
- m: mass of projectile
- v: initial speed at infinity
- b: impact parameter
- θ: scattering angle (deflection angle)
- q1 = Z_p e, q2 = Z_t e
- k = 1/(4πϵ0)
- E = ⁄2 m v^2: kinetic energy
Relation between impact parameter and scattering angle
Classical mechanics relates b and θ for an inverse-square central force. For repulsive Coulomb interaction the scattering angle θ is given by
θ = 2 arctan( (k q1 q2) / (m v^2 b) ).
Define the parameter β = (k q1 q2) / (m v^2 b) = (k q1 q2) / (2E b).
Solving for b as a function of θ:
b(θ) = (k q1 q2) / (m v^2) · cot(θ/2) = (k q1 q2) / (2E) · cot(θ/2).
This monotonic relation between b and θ underlies the mapping from impact-parameter distribution to angular distribution.
Differential cross-section (Rutherford formula)
The differential cross-section dσ/dΩ expresses probability per unit solid angle. For an axially symmetric beam:
dσ = 2π b db, and dΩ = 2π sinθ dθ.
Using conservation of particle flux and b(θ), the differential cross-section follows:
dσ/dΩ = (b / sinθ) |db/dθ|.
Differentiate b(θ) = C cot(θ/2) with C = (k q1 q2)/(m v^2):
db/dθ = C · d/dθ[cot(θ/2)] = C · ( −1/2 csc^2(θ/2) ).
Thus
|db/dθ| = (C/2) csc^2(θ/2).
Compute b/sinθ: b/sinθ = C cot(θ/2) / sinθ.
Using trig identities: sinθ = 2 sin(θ/2) cos(θ/2), cot(θ/2) = cos(θ/2)/sin(θ/2), so
b/sinθ = C [cos(θ/2)/sin(θ/2)] / [2 sin(θ/2) cos(θ/2)] = C / [2 sin^2(θ/2)].
Multiply by |db/dθ|:
dσ/dΩ = (C / [2 sin^2(θ/2)]) · (C/2) csc^2(θ/2) = C^2 / (4 sin^4(θ/2)).
Substitute C:
dσ/dΩ = ( (k q1 q2)^2 / (m^2 v^4) ) · 1 / (4 sin^4(θ/2) ).
Express in terms of kinetic energy E = ⁄2 m v^2:
m^2 v^4 = (2E)^2 = 4 E^2, so
dσ/dΩ = ( (k q1 q2)^2 / (16 E^2) ) · 1 / sin^4(θ/2).
More commonly written as the Rutherford formula:
dσ/dΩ = ( (k q1 q2)^2 / (16 E^2) ) · csc^4(θ/2).
In CGS (Gaussian) units or nuclear/atomic unit conventions the prefactor is adjusted; in many texts for alpha scattering off nucleus of charge Ze this appears as
dσ/dΩ = ( (Z_p Z_t e^2)^2 / (16 (4πϵ0)^2 E^2) ) csc^4(θ/2).
A compact frequently-seen form (using kinetic energy T or projectile momentum p) is
dσ/dΩ = ( (Z_p Z_t e^2) / (8πϵ0 E) )^2 · 1 / sin^4(θ/2).
Total cross-section and divergence
Integrating dσ/dΩ over all solid angle (θ from 0 to π) yields a divergence at small θ because csc^4(θ/2) ~ (4/θ^4) for small θ, so the total cross-section for a pure Coulomb potential is infinite. Physically this reflects the long range of the Coulomb force: arbitrarily distant projectiles are weakly deflected into small angles. Experimental observables use finite angular acceptance or impose a minimum momentum transfer (or include screening) to obtain finite counts.
For a practical finite angular range θ_min to θ_max the integrated cross-section is
σ(θ_min, θmax) = ∫{Ω(θ_min)}^{Ω(θmax)} (dσ/dΩ) dΩ = 2π ∫{θ_min}^{θmax} (dσ/dΩ) sinθ dθ = 2π C^⁄4 ∫{θ_min}^{θ_max} csc^4(θ/2) sinθ dθ.
Using substitution u = θ/2 simplifies the integral; after algebra one obtains a finite expression depending on cot and csc evaluated at the limits.
Angular distributions and characteristic behavior
Key features of the Rutherford angular distribution:
- Strong forward peak: dσ/dΩ ~ θ^{-4} as θ → 0 (since sin(θ/2) ≈ θ/2), producing many small-angle scatterings.
- Symmetric in azimuthal angle φ (axial symmetry).
- Power-law tail: the 1/sin^4(θ/2) dependence means large-angle scattering is rare but measurable for high-Z targets or low-energy projectiles.
Plotting dσ/dΩ vs θ on log-log axes shows a straight-line slope of −4 at small θ.
Quantum mechanical viewpoint
Quantum scattering theory reproduces the Rutherford formula in the first Born approximation for Coulomb scattering at high energies or small coupling. The scattering amplitude f(θ) for a 1/r potential gives |f(θ)|^2 = dσ/dΩ identical to the classical result, though care is needed because the Coulomb potential is a long-range potential and requires regularization (distorted-wave or exact solutions using partial waves lead to the Rutherford result plus a phase). The exact quantum mechanical solution (via the Schrödinger equation for a Coulomb potential) yields the Rutherford cross-section for elastic scattering; interference phases affect only the complex amplitude, not the modulus squared for pure Coulomb.
Quantum considerations introduce criteria for applicability:
- Valid when the de Broglie wavelength λ = h/p is small relative to impact parameters of interest.
- At very small scattering angles (large impact parameters), screening by atomic electrons modifies the effective potential and reduces cross-section.
- At very high energies, relativistic corrections and projectile/nucleus structure (form factors) modify results.
Corrections and limitations
- Screening by atomic electrons: replace bare Coulomb potential with screened potential (e.g., Yukawa-like), which removes forward divergence and modifies small-angle scattering. Molière theory provides multiple-scattering and screening corrections.
- Nuclear size and form factors: for very close approaches (large-angle scattering) the finite nuclear charge distribution causes deviations—introduce nuclear form factor F(q) where q is momentum transfer; dσ/dΩ → dσ_Rutherford/dΩ · |F(q)|^2.
- Relativistic and spin effects: for leptonic projectiles or high energies, use Rutherford–Mott or Mott scattering formulae incorporating spin and relativistic kinematics.
- Multiple scattering: thick targets produce multiple small deflections; Gaussian approximations (Molière, Highland formula) describe net angular spread.
- Inelastic processes: excitation, ionization, or nuclear reactions remove flux from elastic Rutherford channel.
Practical examples and sample calculation
Example: 5 MeV alpha particles (Z_p = 2) scattering off gold nuclei (Z_t = 79). Constants: e^2/(4πϵ0) ≈ 1.44 MeV·fm. Using E = 5 MeV and q1 q2 = (2)(79) e^2:
C = (k q1 q2)/(2E) = (1.44 MeV·fm · 158) / (2·5 MeV) ≈ (227.52 MeV·fm) / 10 MeV ≈ 22.75 fm.
Then dσ/dΩ(θ) = C^2 csc^4(θ/2). At θ = 30° (θ/2 = 15°), sin(15°) ≈ 0.2588 so csc^4(15°) ≈ (3.863)^4 ≈ 222. Multiply by C^2 ≈ (22.75 fm)^2 ≈ 518 fm^2 gives dσ/dΩ ≈ 1.15×10^5 fm^2/sr = 1.15×10^{-25} m^2/sr. (Order-of-magnitude estimate; include unit conversions when needed.)
Experimental considerations
- Measure counts vs angle and compare to Rutherford prediction to extract nuclear charge Z or detect deviations indicating nuclear size or non-Coulomb interactions.
- Use thin, low-Z backing to minimize multiple scattering.
- Use collimation and precise angular detectors to resolve the steep forward peak.
Modern applications
- Particle and nuclear physics: elastic scattering to probe charge distributions (form factors) and search for non-Coulomb interactions.
- Materials analysis: Rutherford Backscattering Spectrometry (RBS) uses backscattered ion energy/angle spectra to determine composition and depth profiles.
- Radiation shielding and beam transport: understanding multiple Coulomb scattering informs beamline design.
Summary
- The Rutherford differential cross-section is dσ/dΩ = ((k q1 q2)^2 / (16 E^2)) csc^4(θ/2), exhibiting a strong forward divergence ~θ−4.
- Total cross-section for an unscreened Coulomb potential diverges; physical setups impose cutoffs via screening, finite detector acceptance, or target thickness.
- Corrections include screening, nuclear form factors, relativistic and spin effects, and multiple scattering; quantum mechanics reproduces the classical result with appropriate treatment.