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.

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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)

Nuclide	Mass number A	Atomic number Z	Element
¹⁴₆C	14	6	Carbon
¹²₆C	12	6	Carbon
⁴₂He	4	2	Helium (α‑particle)

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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 · A^1/3.
• Protons and neutrons are the constituents; electrons belong to the atomic electron cloud and do not affect nuclear reactions directly.

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3. Types of radioactive decay

3.1 α‑decay

Emission of an α‑particle (⁴₂α ≡ ⁴₂He).

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

$$ \,^{A}_{Z}\text{X} \;\rightarrow\; \,^{A}_{Z+1}\text{Y} \;+\; \,^{0}_{-1}\beta^{-} \;+\; \bar{u}_e $$

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

Example

$$ \,^{14}_{6}\text{C} \;\rightarrow\; \,^{14}_{7}\text{N} \;+\; \,^{0}_{-1}\beta^{-} \;+\; \bar{u}_e $$

3.3 β⁺‑decay (positron emission)

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

General equation

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

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

Example

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

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

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

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

Example

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

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

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

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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 t_½: $$t_{½} = \frac{\ln 2}{\lambda}$$
• Activity A (Bq): $$A = \lambda N$$

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5. Mass‑defect and binding energy (introductory)

• The measured mass of a nucleus M_nuc 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).

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6. Summary table of decay modes

Decay mode	Particle(s) emitted	Change in A	Change in Z	Typical example
α‑decay	⁴₂α (He nucleus)	–4	–2	⁸⁸₂U → ⁸⁴₂Th + α
β⁻‑decay	e⁻ (β⁻) + $\bar{u}_e$	0	+1	¹⁴₆C → ¹⁴₇N + β⁻
β⁺‑decay	e⁺ (β⁺) + νe	0	–1	²²₁₁Na → ²²₁₀Ne + β⁺
Electron capture (EC)	νe (no charged particle)	0	–1	⁷₄Be + e⁻ → ⁷₃Li + νe
γ‑decay	γ photon	0	0	⁶⁰₂₇Co* → ⁶⁰₂₇Co + γ

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

Syllabus Requirement	Coverage in these notes	Suggested further activity / emphasis
α‑particle scattering experiment & inference of a small nucleus	Section 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 X	Covered in Section 1 with examples.	None needed.
Conservation of nucleon number & charge in reactions	Explicitly 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 γ	Sections 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, activity	Section 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.

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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: E_b = Δm c², with Δm = Z m_p + (A‑Z) m_n – M_nuc