Discovery of Nucleus (Rutherford)

A radioactive source (radium or radon) emitted alpha particles ($\alpha$-particles), which are helium-4 nuclei: charge +2e, mass ≈ 6.64 × 10⁻²⁷ kg. These particles were collimated into a narrow beam and directed at a gold foil only about 400 atoms thick (~10⁻⁷ m). A circular zinc-sulfide (ZnS) screen surrounded the foil; each impact of an alpha particle on the ZnS screen produced a tiny flash of light (scintillation) that could be detected visually through a microscope. The entire apparatus was evacuated to prevent air from scattering the beam.

Approximately 99.98% of the alpha particles passed through the gold foil with little or no deflection (less than 1°). Under the Thomson plum-pudding model this was not surprising: a spread-out positive charge would exert only weak, gradual electrostatic forces on a passing alpha particle. Rutherford's conclusion: most of the atom is empty space.

A small but measurable fraction of alpha particles (roughly 1 in 10,000) were deflected by angles greater than 90°. Some bounced almost directly backward (θ ≈ 180°). The Thomson model predicted maximum deflections of only a fraction of a degree; it was completely incapable of explaining large-angle scattering. The only way to deflect a fast, massive, doubly-charged alpha particle through a large angle is to have an extremely intense electrostatic repulsive force act over a very short distance.

Rutherford proposed that the atom consists of: (1) a tiny, extremely dense nucleus containing all of the atom's positive charge and almost all of its mass, with a radius of roughly 10⁻¹⁵ m (1 fm); (2) electrons occupying the surrounding space out to a radius of approximately 10⁻¹⁰ m (1 Å). The ratio of nuclear radius to atomic radius is therefore about 1:100,000 — analogous to a marble placed at the centre of a football stadium. An alpha particle that happens to travel close to the nucleus experiences a powerful Coulomb repulsion and is deflected sharply. Those that travel far from the nucleus pass through essentially empty space and are undeflected.

Rutherford derived a mathematical relationship — now called the Rutherford scattering formula — that predicts the number of alpha particles $N(\theta)$ scattered into a detector at angle $\theta$. The formula is: \[ N(\theta) = \frac{N_i \, n \, L \, Z_1^2 Z_2^2 e^4}{16 \, r^2 \, (4\pi\varepsilon_0)^2 \, K^2 \, \sin^4(\theta/2)} \] where $N_i$ is the number of incident particles, $n$ is the number density of target atoms, $L$ is the foil thickness, $Z_1$ and $Z_2$ are the atomic numbers of the projectile and target, $K$ is the kinetic energy of the alpha particle, $r$ is the detector distance, and $e$ is the elementary charge. The $\sin^4(\theta/2)$ dependence was confirmed experimentally by Geiger and Marsden, providing powerful validation of the nuclear model.

For a head-on collision (θ = 180°), the alpha particle momentarily comes to rest as all its kinetic energy converts to electrostatic potential energy. Setting $K = \frac{Z_1 Z_2 e^2}{4\pi\varepsilon_0 d}$ and solving for $d$ gives the distance of closest approach: $d = \frac{Z_1 Z_2 e^2}{4\pi\varepsilon_0 K}$. For gold (Z = 79) and a 7.7 MeV alpha particle: $d \approx \frac{2 \times 79 \times (1.6\times10^{-19})^2}{4\pi(8.85\times10^{-12}) \times 7.7\times10^6\times1.6\times10^{-19}} \approx 3\times10^{-14}$ m = 30 fm. Since the alpha particle does not penetrate the nucleus at this energy, the nuclear radius must be smaller than this value, consistent with modern measurements of ~7 fm for gold.

Rutherford's nuclear model had one major weakness: classical electromagnetism predicts that an electron orbiting a nucleus (an accelerating charge) would continuously radiate energy and spiral into the nucleus within ~10⁻¹¹ s — atoms would not be stable. This problem was resolved in 1913 by Niels Bohr, who introduced quantised electron orbits. Rutherford's nuclear model was also later refined by the discovery of the neutron (James Chadwick, 1932), establishing the nucleus as composed of protons and neutrons (nucleons). Despite these refinements, the core concept — a tiny, dense, positively charged nucleus surrounded by electrons — remains the cornerstone of atomic theory.

• d — Distance of closest approach (m): The minimum separation between the alpha particle and the target nucleus during a head-on (180°) collision, providing an upper bound on nuclear radius.
• Z_1 — Atomic number of projectile (dimensionless): For an alpha particle (helium-4 nucleus), Z₁ = 2.
• Z_2 — Atomic number of target (dimensionless): For gold, Z₂ = 79; for other targets this value changes accordingly.
• e — Elementary charge (C): Magnitude of the charge on one proton or one electron; e = 1.602 × 10⁻¹⁹ C.
• \varepsilon_0 — Permittivity of free space (C² N⁻¹ m⁻²): The electric constant; ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
• K — Kinetic energy of alpha particle (J): Initial kinetic energy of the incident alpha particle; typically 4–9 MeV for naturally emitted alphas (1 MeV = 1.602 × 10⁻¹³ J).

Problem. An alpha particle emitted by radium-226 has a kinetic energy of 4.87 MeV. It is fired directly at a gold nucleus (Z = 79) in a head-on collision. Calculate the distance of closest approach d, and comment on whether the alpha particle penetrates the gold nucleus (nuclear radius of gold ≈ 7.3 fm).

Answer. The distance of closest approach is approximately 46.7 fm (4.67 × 10⁻¹⁴ m). Since this is much larger than the gold nuclear radius (7.3 fm), the alpha particle does not reach the nucleus and is repelled purely by Coulomb electrostatic forces.

• Thomson's Plum-Pudding Model — The plum-pudding model was the prevailing atomic model immediately before Rutherford's experiment; understanding it is essential to appreciate why Rutherford's results were so revolutionary.
• Bohr Model of the Atom — Niels Bohr built directly on Rutherford's nuclear model in 1913, introducing quantised electron orbits to explain atomic stability and emission spectra.
• Protons, Neutrons, and Electrons — Rutherford's nuclear model introduced the concept of a proton-containing nucleus; the later discovery of neutrons (Chadwick, 1932) completed our picture of nuclear composition.
• Atomic Number and Mass Number — The concept of atomic number Z as the number of protons in the nucleus follows directly from Rutherford's nuclear model and Moseley's subsequent X-ray work.
• Radioactivity and Alpha Decay — Alpha particles used in Rutherford's experiment are products of alpha decay; understanding radioactivity explains why alpha sources were available and what properties alpha particles have.