Alpha-Particle Scattering

In 1909, Hans Geiger and Ernest Marsden, working under Ernest Rutherford at the University of Manchester, fired a beam of alpha particles (helium-4 nuclei, charge +2e) at an ultrathin gold foil. According to the then-dominant Thomson 'plum-pudding' model, atoms were diffuse spheres of positive charge with electrons embedded inside — like raisins in a pudding. Such a spread-out charge distribution could only deflect alpha particles by tiny angles, a few degrees at most. What Geiger and Marsden actually observed shattered that picture entirely. Most alpha particles passed straight through the foil as expected, but roughly 1 in 8000 bounced back at angles greater than 90°. Rutherford famously compared it to firing artillery shells at tissue paper and having them bounce back at you. This startling result could only be explained if virtually all of the atom's positive charge and most of its mass were concentrated in an extraordinarily small, dense region — the atomic nucleus. The nucleus has a diameter of roughly 10⁻¹⁵ m (1 femtometre), while the atom itself spans about 10⁻¹⁰ m (1 ångström), making the nucleus about 100,000 times smaller than the atom. Rutherford's nuclear model replaced Thomson's model and laid the foundation for all of modern nuclear and atomic physics. It established that atoms are mostly empty space, that repulsive Coulomb electrostatics can probe nuclear size, and that the distribution of scattered particles encodes the geometry of the scattering centre. Alpha-particle scattering remains one of the most elegant and consequential experiments in the history of science.

Key takeaways
  • The Experimental Setup
  • The Coulomb Repulsion Mechanism
  • The Hyperbolic Trajectory
  • The Rutherford Scattering Formula
  • What the Results Tell Us

The Experimental Setup

A radioactive source (e.g., radon or radium) emits alpha particles that are collimated into a narrow beam. The beam strikes a gold foil approximately 400 nm thick — thin enough that most particles encounter only one or very few gold atoms. A zinc-sulphide (ZnS) detector screen surrounds the foil; each alpha particle that hits the screen produces a tiny flash of light (scintillation) that can be counted. By rotating the detector to different angles θ from the beam direction, Geiger and Marsden measured how many particles arrived at each angle.

The Coulomb Repulsion Mechanism

An alpha particle (charge q₁ = +2e) approaching a gold nucleus (charge q₂ = +79e) experiences a repulsive electrostatic force F = kq₁q₂/r². Unlike a distributed charge cloud, a point-like nucleus exerts a force that grows enormously as the alpha particle gets close (r → 0). For a head-on approach, the alpha particle decelerates to a stop at the distance of closest approach d, where all kinetic energy has been converted to electrostatic potential energy: ½mv² = kq₁q₂/d. This allows an experimental estimate of the nuclear radius upper bound.

The Hyperbolic Trajectory

For any impact parameter b (the perpendicular distance between the initial beam direction and the nucleus), the alpha particle follows a hyperbolic path due to the inverse-square repulsion. A large b means the particle passes far from the nucleus and is deflected only slightly. A small b means a close encounter and a large deflection. The relationship between scattering angle θ and impact parameter b is: b = (kq₁q₂/2E) × cot(θ/2), where E is the kinetic energy of the incoming alpha particle. This is a clean, calculable prediction — no free parameters.

The Rutherford Scattering Formula

Rutherford derived the differential cross-section dσ/dΩ — the effective target area per unit solid angle — for Coulomb scattering. The formula predicts the number of particles N(θ) scattered into a detector at angle θ. The key prediction is that N(θ) ∝ 1/sin⁴(θ/2), meaning far fewer particles scatter to large angles, but the distribution has a definite, measurable shape. Geiger and Marsden's experimental counts matched this formula to remarkable precision, confirming the nuclear model. Where the data deviate from Rutherford's formula (at very high energies or very small angles), physicists can extract information about the nuclear charge distribution itself.

What the Results Tell Us

From the minimum distance of closest approach (head-on collision), the nuclear radius is bounded above. For gold, using typical alpha energies of ~5 MeV, d ≈ 45 fm, much larger than the actual nuclear radius (~7 fm). At higher beam energies, deviations from the Rutherford formula begin to appear, signalling that the alpha particle is actually touching the nuclear surface — providing a direct measurement of nuclear size. This technique of scattering particles to probe internal structure is the ancestor of all modern particle physics experiments, including those at CERN.

The Key Equation

dNdΩ=n t Z12Z22e416Ek2sin⁡4(θ/2)\frac{dN}{d\Omega} = \frac{n\,t\,Z_1^2 Z_2^2 e^4}{16 E_k^2 \sin^4(\theta/2)}
  • dN/d\Omega — Differential count rate (sr⁻¹): Number of particles detected per unit solid angle at angle θ
  • n — Areal number density of target nuclei (m⁻²): Number of gold nuclei per unit area of foil
  • t — Foil thickness (implicit in n) (m): Physical thickness of the target foil
  • Z_1 — Atomic number of projectile (dimensionless): Z₁ = 2 for alpha particle
  • Z_2 — Atomic number of target nucleus (dimensionless): Z₂ = 79 for gold
  • e — Elementary charge (C): e = 1.602 × 10⁻¹⁹ C
  • E_k — Kinetic energy of alpha particle (J): Typically 4–9 MeV for alpha particles from radioactive decay
  • \theta — Scattering angle (radians or degrees): Angle between scattered particle direction and original beam axis

Worked Example

Problem. An alpha particle (charge +2e, mass 6.64 × 10⁻²⁷ kg) with kinetic energy 5.0 MeV is fired directly at a gold nucleus (Z = 79) in a head-on collision. Calculate the distance of closest approach d, where the particle momentarily stops.

d=k q1q2Ek=(8.99×109)(3.20×10−19)(1.27×10−17)8.01×10−13≈4.55×10−14 md = \frac{k\,q_1 q_2}{E_k} = \frac{(8.99\times10^9)(3.20\times10^{-19})(1.27\times10^{-17})}{8.01\times10^{-13}} \approx 4.55\times10^{-14}\text{ m}

Answer. Distance of closest approach ≈ 45.5 fm (4.55 × 10⁻¹⁴ m)

Figure
Diagram
Rutherford Alpha-Particle ScatteringAunucleus (+79e)α (+2e)beam directionsmall deflection(large b)medium deflection180° backscatter(head-on, b≈0)b (impactparameter)Large b → small θ (most particles)Small b → large θ (rare events, reveal nucleus)
Origins · 1 minErnest Rutherford and how this was discovered
ER
Ernest Rutherford
1871–1937 · New Zealand–British

Rutherford designed and interpreted the gold-foil scattering experiment (1909–1911), discovering the atomic nucleus and proposing the nuclear model of the atom. He also discovered the proton, named alpha and beta radiation, and established that radioactive decay transmutes elements — work for which he received the 1908 Nobel Prize in Chemistry. Rutherford was famously pragmatic: he reportedly said 'All science is either physics or stamp collecting.' Despite winning the Nobel Prize in Chemistry, he always considered himself a physicist first.

  • The Nuclear Model of the Atom — Alpha-particle scattering is the direct experimental evidence that led Rutherford to propose the nuclear model, replacing Thomson's plum-pudding model.
  • Coulomb's Law and Electrostatic Force — The repulsive force between the alpha particle and nucleus is purely Coulombic — understanding F = kq₁q₂/r² is essential to derive the scattering trajectories.
  • Alpha Decay and Alpha Particles — Alpha particles are the projectiles in Rutherford's experiment; knowing their charge (+2e) and mass (4u) is necessary to apply the scattering equations.
  • Nuclear Size and Binding Energy — Rutherford scattering gives an upper bound on nuclear radius; deviations from the formula at high energies provide direct measurements of nuclear size.