infer from the results of the α-particle scattering experiment the existence and small size of the nucleus

Objective

To use the results of the α‑particle scattering experiment to infer the existence and extremely small size of the atomic nucleus, and to integrate this evidence with the wider Cambridge IGCSE/A‑Level syllabus on atoms, nuclei, radiation, fundamental particles and nuclear energetics.

1. Nuclear Notation, Isotopes & Conservation Laws

2. Fundamental Particles

• Quarks – six flavours (up, down, charm, strange, top, bottom).

  • Up (u): charge +2⁄3 e
  • Down (d): charge –1⁄3 e

  Protons = uud (charge +1 e); neutrons = udd (charge 0 e).

• Baryons – particles made of three quarks (e.g. p, n).
• Mesons – quark–antiquark pairs (e.g. π⁺, π⁰, K⁻).
• Leptons – elementary particles that do not experience the strong force. Relevant families for the syllabus:

  • Electron (e⁻) and electron‑antineutrino ( \(\bar{\nu}_e\) )
  • Positron (e⁺) and electron‑neutrino ( \(\nu_e\) )

• Antiparticles – each particle has a counterpart with opposite charge (e⁺ is the antiparticle of e⁻, \(\bar{\nu}e\) of \(\nue\), etc.).
• β‑decay in quark language

  • β⁻ decay: d → u + e⁻ + \(\bar{\nu}_e\)
  • β⁺ decay (or electron capture): u → d + e⁺ + \(\nue\) (or p + e⁻ → n + \(\nue\)).

3. Types of Nuclear Radiation

4. Radioactive Decay Equations

• α‑decay \(\displaystyle \,^{A}{Z}\!X \;\rightarrow\; ^{A-4}{Z-2}\!Y \;+\; ^{4}_{2}\!He\)
• β⁻‑decay \(\displaystyle \,^{A}{Z}\!X \;\rightarrow\; ^{A}{Z+1}\!Y \;+\; e^- \;+\; \bar{\nu}_e\)
• β⁺‑decay \(\displaystyle \,^{A}{Z}\!X \;\rightarrow\; ^{A}{Z-1}\!Y \;+\; e^+ \;+\; \nu_e\)
• Electron capture (EC) \(\displaystyle \,^{A}{Z}\!X \;+\; e^- \;\rightarrow\; ^{A}{Z-1}\!Y \;+\; \nu_e\)
• γ‑emission (de‑excitation) \(\displaystyle \,^{A}{Z}\!X^{*} \;\rightarrow\; ^{A}{Z}\!X \;+\; \gamma\)

5. Radioactive‑Decay Kinetics

• Activity (A) – number of decays per unit time (Bq).
• Decay constant (λ) – probability per unit time that a nucleus will decay.
• Relationship: \(A = \lambda N\) (where N = number of undecayed nuclei).
• Half‑life (t½) – time for N to fall to N⁄2. \(t_{½} = \dfrac{\ln 2}{\lambda}\).
• Exponential decay law \(N(t) = N0 e^{-\lambda t}\) and \(A(t)=A0 e^{-\lambda t}\).

6. Mass Defect, Binding Energy & the Nuclear Binding‑Energy Curve

• Mass defect (Δm) – difference between the sum of the masses of the separate nucleons and the actual nuclear mass:

  \(\displaystyle \Delta m = \bigl(Z mp + (A-Z) mn\bigr) - m_{\text{nucleus}}\).

• Binding energy (E_b) – energy required to separate a nucleus into its constituent nucleons:

  \(\displaystyle E_b = \Delta m \,c^{2}\) (1 u ≈ 931.5 MeV).

• Binding‑energy per nucleon – \(E_b/A\). The curve rises rapidly for light nuclei, peaks around A ≈ 56 (Fe‑56), then falls slowly for heavier nuclei. This explains why:

  • Fusion of very light nuclei releases energy (moving up the curve).
  • Fission of very heavy nuclei releases energy (moving down the curve).

Qualitative sketch (textual description): a bell‑shaped curve starting near 0 MeV / nucleon at A = 1, rising to ≈ 8.8 MeV / nucleon at A ≈ 56, then slowly decreasing to ≈ 7.6 MeV / nucleon for A ≈ 240.

Example calculations

• Fusion – Deuterium + Tritium:

  \(\displaystyle ^{2}{1}\!H + ^{3}{1}\!H \;\rightarrow\; ^{4}{2}\!He + ^{1}{0}\!n\)

  Δm ≈ 0.018 u → Q ≈ 17.6 MeV.

• Fission – Uranium‑235 thermal fission:

  \(\displaystyle ^{235}{92}\!U + ^{1}{0}\!n \;\rightarrow\; ^{90}{36}\!Kr + ^{143}{56}\!Ba + 3\,^{1}_{0}\!n\)

  Δm ≈ 0.186 u → Q ≈ 173 MeV.

7. The Rutherford (Geiger–Marsden) α‑Particle Scattering Experiment

7.1 Experimental Arrangement

• α‑particle source (He²⁺) placed in a high‑vacuum chamber.
• Collimator produces a narrow, well‑defined beam.
• Beam strikes a thin metal foil (typically gold, thickness ≈ 10⁻⁸ m).
• Surrounding circular ZnS scintillation screen; a microscope counts scintillations at various angles θ.

7.2 Key Observations

1. ~ 98 % of α‑particles pass straight through the foil (θ ≈ 0°).
2. A small fraction are deflected through large angles; a few are reflected backwards (θ ≈ 180°).
3. The number of particles scattered at a given angle falls off sharply with increasing θ.

7.3 Why the Thomson “Plum‑Pudding” Model Fails

If the positive charge were uniformly spread throughout the atom, the internal electric field would be weak and α‑particles would experience only gentle, continuous deflection. The observed large‑angle scattering is impossible under that model.

7.4 Rutherford’s Nuclear Model

• The atom contains a tiny, massive, positively charged nucleus.
• All of the atomic mass and positive charge are concentrated in the nucleus.
• Electrons occupy the surrounding space and have negligible effect on the scattering of heavy α‑particles.

7.5 Rutherford Scattering Formula

The differential cross‑section for Coulomb scattering of a particle of charge \(Z{1}e\) by a nucleus of charge \(Z{2}e\) is

\[

\frac{d\sigma}{d\Omega}= \left(\frac{Z{1} Z{2} e^{2}}{16\pi\varepsilon_{0}E}\right)^{2}\frac{1}{\sin^{4}(\theta/2)}

\]

where \(E\) is the kinetic energy of the incident α‑particle and θ the scattering angle.

7.6 Estimating the Size of the Nucleus

Impact parameter

The impact parameter \(b\) is the perpendicular distance between the initial α‑particle trajectory and the centre of the nucleus. Large‑angle scattering requires \(b\) to be comparable to the nuclear radius \(R\).

Back‑scattered α‑particle (θ ≈ 180°)

Energy conservation gives the distance of closest approach \(r_{\min}\):

\[

\frac{1}{4\pi\varepsilon{0}}\frac{Z{1} Z{2} e^{2}}{r{\min}} = E

\quad\Longrightarrow\quad

r{\min}= \frac{Z{1} Z{2} e^{2}}{4\pi\varepsilon{0}E}

\]

Taking a typical gold foil experiment:

• \(E \approx 5\;\text{MeV}\)
• \(Z_{1}=2\) (α‑particle)
• \(Z_{2}=79\) (gold)

\[

R \lesssim r_{\min}= \frac{(2)(79)(1.60\times10^{-19}\,\text{C})^{2}}

{4\pi(8.85\times10^{-12}\,\text{F m}^{-1})(5\times10^{6}\,\text{eV})(1.60\times10^{-19}\,\text{J eV}^{-1})}

\approx 1.5\times10^{-14}\,\text{m}

\]

This is roughly 10 000 times smaller than a typical atomic radius (≈ 10⁻¹⁰ m), confirming the nucleus’s extreme compactness.

Empirical radius formula

\[

R = r{0}A^{1/3}, \qquad r{0}\approx 1.2\;\text{fm}\;(1\;\text{fm}=10^{-15}\,\text{m})

\]

The relation fits the order‑of‑magnitude estimate from Rutherford scattering for all known nuclei.

8. Representative Nuclear Reactions (including fission & fusion)

9. Summary Tables

9.1 Radiation Properties (quick revision)

9.2 Nuclear vs Atomic Size

9.3 Binding‑Energy per Nucleon (qualitative)

Peak ≈ 8.8 MeV / nucleon at A ≈ 56 (Fe‑56); decreases slowly for heavier nuclei, explaining the energy release in both fusion (light nuclei) and fission (heavy nuclei).

10. Conclusions

• The α‑particle scattering experiment provides direct, quantitative evidence that atoms contain a tiny, dense, positively charged nucleus.
• ~ 98 % of α‑particles pass straight through because the nucleus occupies only ~10⁻⁴ of the atomic volume.
• Large‑angle scattering shows that the nuclear charge is confined to a region of radius 10⁻¹⁴–10⁻¹⁵ m, about ten thousand times smaller than the atom itself.
• Understanding nuclear notation, isotopes, fundamental particles, decay kinetics, mass‑energy relations and binding‑energy trends completes the Cambridge 9702 (IGCSE) and A‑Level syllabus, linking the scattering experiment to the broader study of nuclear reactions, radiation detection and nuclear energy.