represent α- and β-decay by a radioactive decay equation of the form UT h92238 90234

Atoms, Nuclei and Radiation – A‑Level Physics (Cambridge 9702)

Learning Objective

Students will be able to:

  • Write nuclear decay equations for α‑, β⁻‑, β⁺‑, electron‑capture (EC) and γ‑decay using correct nuclear notation.
  • Explain how the mass number A and atomic number Z change in each decay mode, invoking the conservation of nucleon number and electric charge.
  • Describe the experimental evidence for a small, positively‑charged nucleus (Rutherford scattering) and the simple nuclear model used in the syllabus.
  • Apply the radioactive decay law, half‑life, activity and basic concepts of mass‑defect and binding energy.


1. Nuclear notation and basic concepts

  • General form: ⁽ᴬ⁾₍ᶻ₎X

    • A – mass number = total number of nucleons (protons + neutrons).
    • Z – atomic number = number of protons (defines the element).
    • X – chemical symbol of the element.

  • Isotopes: Same Z, different A.


    Example: ¹⁴₆C (6 p, 8 n) vs ¹²₆C (6 p, 6 n).

  • Nuclide: A specific combination of A and Z (e.g. ²³⁸₉₂U).
  • Conservation laws in nuclear reactions

    • Nucleon number (A) is conserved – the total number of protons + neutrons before and after a reaction is the same.
    • Electric charge (Z) is conserved – the total proton charge before and after a reaction is the same.

1.1 Distinguishing A and Z (side‑by‑side example)

NuclideMass number AAtomic number ZElement
¹⁴₆C146Carbon
¹²₆C126Carbon
⁴₂He42Helium (α‑particle)


2. Experimental evidence for a small nucleus

2.1 Rutherford α‑particle scattering experiment

  • Thin gold foil was bombarded with fast α‑particles (He nuclei).
  • Most α‑particles passed through with little deflection, but a small fraction were scattered at large angles.
  • Interpretation: a tiny, dense, positively‑charged nucleus occupies only ~10⁻¹⁵ m of the atom, while electrons occupy the remaining volume.

Key inference for the syllabus: the atom consists of a central nucleus (protons + neutrons) surrounded by orbiting electrons.

2.2 Simple nuclear model (ball model)

  • The nucleus is treated as a compact sphere of radius R ≈ 1.2 fm · A1/3.
  • Protons and neutrons are the constituents; electrons belong to the atomic electron cloud and do not affect nuclear reactions directly.


3. Types of radioactive decay

3.1 α‑decay

Emission of an α‑particle (⁴₂α⁴₂He).

General equation

\$\$

\,^{A}{Z}\text{X} \;\rightarrow\; \,^{A-4}{Z-2}\text{Y} \;+\; \,^{4}_{2}\alpha

\$\$

  • Mass number decreases by 4.
  • Atomic number decreases by 2.

Example

\$\$

\,^{238}{92}\text{U} \;\rightarrow\; \,^{234}{90}\text{Th} \;+\; \,^{4}_{2}\alpha

\$\$

3.2 β⁻‑decay (electron emission)

A neutron transforms into a proton, emitting an electron and an antineutrino (often omitted in syllabus work).

General equation

\$\$

\,^{A}{Z}\text{X} \;\rightarrow\; \,^{A}{Z+1}\text{Y} \;+\; \,^{0}{-1}\beta^{-} \;+\; \bar{\nu}e

\$\$

  • Mass number unchanged.
  • Atomic number increases by 1.

Example

\$\$

\,^{14}{6}\text{C} \;\rightarrow\; \,^{14}{7}\text{N} \;+\; \,^{0}{-1}\beta^{-} \;+\; \bar{\nu}e

\$\$

3.3 β⁺‑decay (positron emission)

A proton converts into a neutron, emitting a positron and a neutrino.

General equation

\$\$

\,^{A}{Z}\text{X} \;\rightarrow\; \,^{A}{Z-1}\text{Y} \;+\; \,^{0}{+1}\beta^{+} \;+\; \nue

\$\$

  • Mass number unchanged.
  • Atomic number decreases by 1.

Example

\$\$

\,^{22}{11}\text{Na} \;\rightarrow\; \,^{22}{10}\text{Ne} \;+\; \,^{0}{+1}\beta^{+} \;+\; \nue

\$\$

3.4 Electron capture (EC)

An inner‑shell electron is captured by the nucleus, turning a proton into a neutron and emitting a neutrino.

General equation

\$\$

\,^{A}{Z}\text{X} \;+\; e^{-} \;\rightarrow\; \,^{A}{Z-1}\text{Y} \;+\; \nu_e

\$\$

  • Mass number unchanged.
  • Atomic number decreases by 1 (same net effect as β⁺‑decay).

Example

\$\$

\,^{7}{4}\text{Be} \;+\; e^{-} \;\rightarrow\; \,^{7}{3}\text{Li} \;+\; \nu_e

\$\$

3.5 γ‑decay (photon emission)

An excited nucleus releases excess energy as a high‑energy photon.

General equation

\$\$

\,^{A}{Z}\text{X}^{*} \;\rightarrow\; \,^{A}{Z}\text{X} \;+\; \gamma

\$\$

  • No change in A or Z; only the nuclear energy state changes.

Example

\$\$

\,^{60}{27}\text{Co}^{*} \;\rightarrow\; \,^{60}{27}\text{Co} \;+\; \gamma

\$\$


4. Radioactive decay law

  • Number of undecayed nuclei after time t:

    \$N = N_{0}\,e^{-\lambda t}\$

  • Decay constant λ (s⁻¹): probability per unit time that a given nucleus decays.
  • Half‑life t½:

    \$t_{½} = \frac{\ln 2}{\lambda}\$

  • Activity A (Bq):

    \$A = \lambda N\$


5. Mass‑defect and binding energy (introductory)

  • The measured mass of a nucleus Mnuc is less than the sum of the masses of its constituent protons and neutrons.
  • Mass‑defect:

    \$\Delta m = Z\,m{p} + (A-Z)\,m{n} - M_{\text{nuc}}\$

  • Binding energy:

    \$E_{b} = \Delta m\,c^{2}\;\;\;( \approx 931.5\;\text{MeV per atomic mass unit})\$

  • Higher binding energy per nucleon ⇒ greater nuclear stability (maximum near ⁵⁶₍₂₆₎Fe).


6. Summary table of decay modes

Decay modeParticle(s) emittedChange in AChange in ZTypical example
α‑decay⁴₂α (He nucleus)–4–2⁸⁸₂U → ⁸⁴₂Th + α
β⁻‑decaye⁻ (β⁻) + \$\bar{\nu}_e\$0+1¹⁴₆C → ¹⁴₇N + β⁻
β⁺‑decaye⁺ (β⁺) + νe0–1²²₁₁Na → ²²₁₀Ne + β⁺
Electron capture (EC)νe (no charged particle)0–1⁷₄Be + e⁻ → ⁷₃Li + νe
γ‑decayγ photon00⁶⁰₂₇Co* → ⁶⁰₂₇Co + γ


7. Alignment with Cambridge 9702 syllabus (Topics 11.1 & 11.2)

Syllabus RequirementCoverage in these notesSuggested further activity / emphasis
α‑particle scattering experiment & inference of a small nucleusSection 2.1 gives a concise description and the key inference.Include a labelled diagram of the gold‑foil experiment for visual learners.
Simple nuclear model (protons, neutrons, orbital electrons)Section 2.2 outlines the ball model and distinguishes nuclear from electronic structure.Add a schematic “nucleus + electron cloud” illustration.
Distinguish nucleon number (A) from proton number (Z)Table 1.1 (side‑by‑side example) reinforces the distinction.Use a short classroom quiz: “Identify A and Z for ⁶⁴₈Gd”.
Isotopes & notation A Z XCovered in Section 1 with examples.None needed.
Conservation of nucleon number & charge in reactionsExplicitly stated in Section 1 and applied in each decay equation.Add a worked example showing how to balance a mixed‑decay reaction.
Write decay equations for α, β⁻, β⁺, EC and γSections 3.1–3.5 provide general equations and worked examples.Practice worksheet: complete the missing products for given parent nuclides.
Radioactive decay law, half‑life, activitySection 4 presents the formulas with definitions.Include a sample calculation of the activity of a 1 g sample of ⁶⁰Co.
Mass‑defect and binding energy (introductory)Section 5 introduces the concepts and key equations.Demonstrate a simple calculation for the binding energy of ⁴He.


8. Quick reference cheat‑sheet (for revision)

  • α‑decay: ⁽ᴬ⁾₍ᶻ₎X → ⁽ᴬ⁻⁴⁾₍ᶻ⁻²₎Y + ⁴₂α
  • β⁻‑decay: ⁽ᴬ⁾₍ᶻ₎X → ⁽ᴬ⁾₍ᶻ⁺¹₎Y + ⁰₋₁β⁻ (+ ν̅ₑ)
  • β⁺‑decay: ⁽ᴬ⁾₍ᶻ₎X → ⁽ᴬ⁾₍ᶻ⁻₁₎Y + ⁰₊₁β⁺ (+ νₑ)
  • Electron capture: ⁽ᴬ⁾₍ᶻ₎X + e⁻ → ⁽ᴬ⁾₍ᶻ⁻₁₎Y + νₑ
  • γ‑decay: ⁽ᴬ⁾₍ᶻ₎X* → ⁽ᴬ⁾₍ᶻ₎X + γ
  • Decay law: N = N₀ e⁻ˡᵃᵐᵇᵈᵃ t, t½ = ln 2 / λ, A = λN
  • Binding energy: Eb = Δm c², with Δm = Z mp + (A‑Z) mn – Mnuc