Describe how the scattering of alpha (α) particles by a sheet of thin metal supports the nuclear model of the atom, by providing evidence for: (a) a very small nucleus surrounded by mostly empty space (b) a nucleus containing most of the mass of the

5.1.1 The Atom – Rutherford Scattering Experiment

1. The nuclear model of the atom

  • The atom consists of a tiny, positively‑charged nucleus that contains ≈ 99.9 % of the atom’s mass and all of its positive charge.

  • Electrons form a cloud that occupies a volume about 10⁵ times larger in radius than the nucleus.

  • Consequently an atom is mostly empty space and the mass is essentially concentrated in the nucleus.

2. Formation of ions (relevant to the nuclear model)

Removing one or more electrons produces a positive ion; adding electrons produces a negative ion.

Because the massive nucleus remains unchanged, ion formation involves only the electrons.

3. Rutherford (Geiger–Marsden) scattering experiment

3.1 Set‑up (qualitative description)

  • Source: a sealed α‑particle emitter (e.g. 238Pu) that releases He²⁺ nuclei.

  • Collimation: a narrow aperture creates a thin, well‑defined beam of α‑particles.

  • Target: an ultra‑thin sheet of gold foil (≈ 0.1 µm thick).

  • Detector: a circular fluorescent screen (or a series of Geiger‑Müller tubes) surrounding the foil, marked with angular divisions.

  • Safety note: α‑sources are sealed in metal containers and handled with lead shielding; the beam is confined to the bench and the foil is never inhaled or touched.

3.2 Observations (percentage of α‑particles detected at different angles)

Deflection angle≈ % of α‑particlesWhat this tells us
0° (straight through)≈ 96 %Most of the atom is empty; particles pass without encountering a strong electric field.
0° – 30° (small‑angle scattering)≈ 4 %Weak repulsion from the diffuse electron cloud.
> 90° (large‑angle scattering)≈ 0.5 %Direct encounter with a very small, massive, positively‑charged nucleus.

3.3 How the results support the nuclear model

  1. Very small nucleus surrounded by mostly empty space

    • ≈ 96 % of α‑particles go straight through → the dense region occupies only a tiny fraction of the atomic volume.

    • The rare large‑angle deflections show that a small, localized centre (the nucleus) must exist.

  2. The nucleus contains most of the atom’s mass

    • To change the momentum of a fast α‑particle (mass ≈ 4 u, speed ≈ 2 × 10⁷ m s⁻¹) a comparably massive target is required.

    • Only a massive particle – the nucleus – can produce the observed sharp reversals; electrons are far too light.

  3. The nucleus is positively charged

    • α‑particles carry a +2e charge. Their repulsion (deflection away from the centre) proves that the nucleus also carries a positive charge.

  4. Refutation of the Thomson “plum‑pudding” model

    • The Thomson model predicted that α‑particles would be deflected only slightly, because the positive charge was thought to be spread uniformly throughout the atom.

    • The observation of a few particles being reflected back at angles > 90° is impossible under that model, confirming the existence of a compact nucleus.

3.4 Optional/extended – key equations (for A‑Level or interested students)

Show equations

  • Scattering angle (Rutherford formula)

    \[

    \theta = 2\arctan\!\left(\frac{Z e^{2}}{4\pi\varepsilon_{0}\,m v^{2}\,b}\right)

    \]

  • Differential cross‑section (probability of scattering into a solid angle \(d\Omega\))

    \[

    \frac{d\sigma}{d\Omega}= \left(\frac{Z e^{2}}{8\pi\varepsilon_{0} m v^{2}}\right)^{2}\frac{1}{\sin^{4}(\theta/2)}

    \]

These formulas reproduce the observed distribution when Z equals the atomic number of the foil (gold = 79).

4. Glossary of symbols (quick reference)

SymbolMeaning
\( \alpha \)alpha particle (He²⁺ nucleus)
\( Z \)atomic number of the target atom (number of protons)
\( e \)elementary charge (1.60 × 10⁻¹⁹ C)
\( \varepsilon_{0} \)vacuum permittivity (8.85 × 10⁻¹² C² N⁻¹ m⁻²)
\( m \)mass of the α‑particle (≈ 4 u)
\( v \)speed of the α‑particle (≈ 2 × 10⁷ m s⁻¹)
\( b \)impact parameter – the perpendicular distance between the particle’s initial path and the centre of the nucleus (if no deflection occurs)
\( \theta \)deflection (scattering) angle

5. Conceptual summary

  • The overwhelming majority of α‑particles pass straight through the foil → atoms are mostly empty space.

  • A tiny fraction are sharply deflected, some even back‑scattered → a small, dense, positively‑charged nucleus exists.

  • The nucleus carries almost all the mass of the atom, explaining why only a massive target can change the α‑particle’s momentum.

  • These observations directly contradict the Thomson “plum‑pudding” model and form the experimental foundation of the nuclear model of the atom, which underlies later theories such as the Bohr model and quantum mechanics.

Suggested diagram: schematic of the Rutherford scattering set‑up showing the α‑particle source, thin gold foil, and surrounding fluorescent detection screen with angles marked.